WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about piecewise functions.
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So far, all of the functions we have seen use a single rule over their entire domain.
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No matter what goes into the function, as long as it is part of the domain, the same process happens to that input.
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For example, consider x² + 3; the same process happens, no matter what input goes in.
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While f(1), f(27), f(-473) all produce different outputs, the function is doing the exact same thing to each.
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It squares the input; then it adds 2; f(1) is 1² + 3; f(27) is 27² + 3; f(-473) is (-473)² + 3.
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Different inputs will make different outputs, but ultimately the process we are going through is the same for each one of these.
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What if the rule wasn't always the same, though--what if the rule we used, the process we used, changed depending on the input?
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Instead of always using the same process, the transformation could vary based on what goes into the function.
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This is the idea behind piecewise functions--functions that do different things over different pieces of their domain.
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Let's start by looking at two analogies, to get this idea really into our heads.
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The first analogy: imagine you are cooking potatoes, and you are going to turn them either into mashed potatoes or French fries.
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Now, since the large potatoes make better French fries, you decide to turn your big potatoes into fries, and to mash up the small potatoes.
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You just bought a big pile of potatoes, so you have some big potatoes and some small potatoes.
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So, some of them are going to make better fries, and some of them would probably make better mashed potatoes.
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So, the big ones become fries, and the small ones will become mashed potatoes--OK, that makes sense.
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When you start cooking, you begin to work through your pile of potatoes.
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You pick up one potato at a time, and you start by deciding if it is small or if it is big.
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If it is a small potato, you boil it and mash the potato; if it is a big potato, you cut it up and turn it into fries--it seems pretty simple.
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But we are getting sort of that idea of piecewise functions here, because in both cases, something will happen to the potato.
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The potato will go through a process, but the transformation is different, depending on the type.
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Depending on what kind of potato we have--if it is big potato or if it is a small potato--different processes are going to occur next.
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We have to pay attention to what we are putting into our cooking process before we know what steps to take next.
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Another one--here is another analogy: let's consider a factory.
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Imagine this factory where, depending on what materials you bring to the factory, it produces different things.
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If you bring along two kilograms of wood, they make a chair.
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If you bring in 400 kilograms of metal, they make a car.
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2 kilograms of wood--the factory makes a chair; if you bring in 400 kilograms of metal to the factory, it makes a car.
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Now, notice that it is not enough to simply say how much material you bring.
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If you bring in a mass of 400 kilograms, that is not enough information.
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We need to know if it is wood or metal; if it is wood, then 400 kilograms of wood at 2 kilograms for a chair means we will make 200 chairs.
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But if it is 400 kilograms of metal, well, it was 400 kilograms of metal to make 1 car; so you would get one car.
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400 kilograms isn't enough information; we need to know 400 kilograms and what type it is.
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We need to know what category it belongs to.
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So, something is going to happen to the material, no matter what; but we have to know the category that the material belongs to.
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We have to know what we are putting in--not just a specific number, but what sort of group it is from, before we can tell what is going to happen.
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This is now a vague sense of piecewise functions; we have this idea that a piecewise function
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is something where a process will happen to the input, but different things will happen, depending on the specific nature of the input.
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That is a really good sense for what a piecewise function is.
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So now, let's consider the notation that is used for piecewise functions.
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Generally, it is f(x) = [...and this bracket just says that it breaks into multiple different things.
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There are different possible paths that we can take.
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So, transformation rule 1 is our normal rule, like x² + 7; and then it says x is in category 1.
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So, what that means is that we look at our input; and then we go and we look at the various categories we have.
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Is x wood? Is x metal? Is the potato big? Is the potato small?
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We look for which category it belongs to; once we have found that it belongs to category 2,
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then we go ahead and use transformation rule 2 on that input.
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We look at the input that is going into the function; we then see which one of the categories it belongs to.
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And that tells us which of the rules to use.
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When we want to talk about the function, we have this bracket, so we can see all of the possible rules at once,
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and all of the categories that go along with the possible rules.
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Which set of circumstances do we use each rule under?
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Given some input, we first check which category it belongs to.
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Then, we use the corresponding transformation; so there are all of these transformation rules; we first check the category; then we use it.
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Also, notice that since f is a function, two categories cannot overlap.
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So, categories cannot overlap--why? because, if they overlapped, and they had different rules,
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we would get two different outputs from using the same input.
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We would get two different outputs from using two different rules, if the categories overlapped.
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Remember, if we put x into a function, it has to only produce one output.
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If we put in x into a function, it can't put out a and put out b; it is not allowed to produce two different things.
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So, if we put in x, and x belongs to two different categories, each of those rules would have to either be the same thing,
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or we would have to make sure that the categories don't overlap.
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We are allowed to have categories overlap; but if that is the case, we have to make sure that the transformation rules
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produce the same output during that overlapping space; otherwise, we have broken the nature of being a function.
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All right, let's start looking at some examples to help us get a sense of how to use this notation.
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So, this isn't really formal mathematics; but we can get an idea of how this notation works by seeing how it would work on those previous analogies.
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First, our potatoes analogy: potatoes of input...how does our potato function work on input?
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If we plug in x, the first thing we do is see, "Is x small?"
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If x is small, then we turn it into mashed potatoes; if x is big, then we turn it into French fries.
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So, we plug in our potato x, and then we see which category x belongs to.
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The same sort of thing is going on at the factory.
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If we plug x, our input, into the factory function, some sort of quantity--some number of kilograms--of a material,
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we then say, "OK, is x wood? If x is wood, it is x/2," because remember, it took 2 kilograms of wood to make one chair.
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So, it is x/2 chairs; or if x happens to be metal, it is x/400 cars, because it was 400 kilograms of metal to make one car.
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We plug in the x; we take the x; we see which category it belongs to; and then we plug it into the appropriate rule, based on the category.
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Great; all right, let's see an example of the piecewise function, actually working through with numbers.
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Here is a table; this is the most extreme sort of table we can use.
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When we are actually doing this, we probably won't want to use a table that has this much possible information in it.
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But we will get the idea of how piecewise functions work from this table.
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So, to start with, let's look at what would happen to -4.
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Well, actually, first let's look...which one would -4 belong to?
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-4 would be in x < -1; so it is going to belong in x² - 1, and it is not going to belong in the 2 rule,
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because -4 is not between -1 and 1, and -4 is not greater than 1.
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So, it is going to knock out these two rules.
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Next, -3: -3 is still less than -1, so once again, that knocks out the second rule and the third rule.
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What about -2? Well, once again, -2 is still less than -2, so that knocks out the second rule and the third rule again.
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What about if we plug in -1? Well, -1 is not less than -1; -1 equals -1, so we have this less than or equal right here.
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-1 is less than or equal; it knocks out that first rule, but -1 is still not greater than 1 from our third category.
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So, it knocks out the third rule, as well.
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What about plugging in 0? Well, 0 is not less than -1; and 0 is not greater than 1;
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so our first and third categories just got knocked out--the first and third rules are out.
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What if we plug in 1? Well, once again, 1 is equal to 1, so it is part of this second category.
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1 is not less than -1, so it knocks out the first rule; and 1 is not greater than 1, so it knocks out the third rule.
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We get to 2, and finally 1 is less than 2; 2 is greater than 1; we have 1 being less than 2,
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so we are using the third rule, which means that our first rule and our second rule are knocked out.
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What about 3? 3 is still greater than 1; 3 is not less than -1, and 3 is not between -1 and 1, so those rules and categories are out.
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What about 4? 4 is not less than -1, and 4 is not between -1 and 1; but 4 is greater than 1, so only the third rule gets used there.
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So now, we have a sense of how this table comes together.
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So now, let's actually start plugging in numbers.
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-4 goes into x² - 1: (-4)² - 1 gets us +16 - 1, so we get 15.
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What about -3? We plug in (-3)² - 1; that gets 9 - 1, so we get 8.
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What about -2? We plug in (-2)² - 1; 4 - 1 gets us 3.
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All right, now we switch rules; for this one, we plug in -1, but -1 doesn't really do anything.
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All the function says is that, if you are between -1 and 1 as your input, it outputs 2.
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It doesn't care what you are putting in as an input, as long as it is between -1 and 1.
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It is going to be constant; it is going to always give the same thing in there; so it is going to just be 2, 2, 2; 2 for all of those.
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-1, 0, and 1; it is 2...it is going to be a constant value of 2 in that interval.
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Now, we switch rules once again, and we are at 2, -2(2) + 4; -2(2) gets us -4; -4 + 4 gets us 0.
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What about 3? -2...plug in our 3...+ 4; -2(3) gets us -6; -6 + 4 gets us -2.
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Plug in 4: -2(4) + 4...-2 times 4 gets us -8; -8 + 4 gets us -4.
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We have managed to fill out this table.
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The important thing is to start by figuring out which one of these inputs is going to go to which category.
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Where are my inputs going to go? You have to figure out an input and its connection to which of the possible categories it can be connected to.
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All right, let's also see an example of a non-numerical piecewise.
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Many lessons ago, when we first introduced the idea of a function, we talked about a non-numerical initial function.
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It took in names spelled with the Roman alphabet, and it output the first letter of the name.
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For example, if we gave it the name Robert, the initial function would come along,
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and it would say, "Your first letter is R, so we put out the letter R; done!"
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So, it is just going and saying, "Let's grab the first initial and do that."
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That was our idea of the initial function when we first introduced it.
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We can have functions operating on non-numerical things.
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But we can also have piecewise functions on non-numerical things.
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We can modify that and make a piecewise function; we will have f(x) is equal to two categories.
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Our first rule will be the first letter of x, the first letter of the name, if the name starts with A to M (x is just a placeholder for a name here).
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It is the first letter of the name, if x starts with A - M.
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And then, it is the last letter of the name, if the name starts with N - Z.
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And notice that that covers all of the possible letters that names could start with: A to M, N to Z;
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A, B, C, D, E, F, G, H, I, J, K, L, M; N, O, P, Q, R, S, T, U, V, W, X, Y, Z; great.
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Albert: we plug in Albert, and Albert belongs to the red category; it belongs to starting with A to M.
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That one is pretty easy; we use the first letter, so it gets A as the letter out of it.
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What about Isabella? Well, Isabella is between A and M (A, B, C, D, E, F, G, H, I, J, K, L, M);
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so Isabella also belongs to the red category; so it is going to return an I.
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What about Nicole? Nicole is an N, so it is using the blue category; so it uses the last letter of the name.
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The last letter of Nicole is E.
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Vincent begins with a V, which is between N and Z (M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z).
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So, we are going to be using the blue category, the second category where you use that last letter, T.
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What about Zach? If we have Zach, Zach is pretty clearly going to be starting with a Z.
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That means we are in the second category; we are going to cut off that last letter.
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We are going to get the last letter, and it will be H.
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We can have a piecewise function operating on non-numerical things.
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The idea of a piecewise function is just that we take in things; we see which category the thing belongs to;
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and then we apply a rule based on the category it belongs to.
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We figure out the category; then we apply a rule based on it; great.
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Graphing piecewise functions: how do we graph these things?
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It is actually really similar to graphing a normal function.
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A series of points, (x,f(x)): when we graph x², the reason why (0,0) is a point
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is because, if we plug in 0, we get 0; (0,f(0)); for x², we get a (0,0).
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Then we plug in (1,f(1)), because we plug in 1, and it gets 1²; so it is (1,1).
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We plug in 2, and we get f(2); so that would be 2; and then 2² is 4, so we would be at (2,4).
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We are moving to a height of 4; and that is how it is working for x².
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And that is how it works for graphing any function; we put in the input, and then we see what output that gets to.
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The difference with a piecewise function is that the rule determining where x is going to map will change, depending on which x we are looking at.
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So, often it will look to use like the graph is changing at switchovers--that we are breaking from one thing to another.
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And in a way, we are: we are switching between rules.
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So, when x switches from one category to another, the shape or location of the curve can change,
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because all of a sudden, we are doing a new way of outputting things.
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It is not that nice, smooth connection anymore, because we are working all through x².
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All of a sudden, we are jumping from x² to maybe x³ - 1.
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And we are going to see, suddenly, a new, totally different kind of output when we change categories.
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An important graphical note is that, to show inclusion, we use a solid circle.
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Solid circles say "this point is here--this point is actually being included."
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We show exclusion, excluding it, saying it is not there, with an empty circle.
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Empty circles give us exclusion; exclusion is an empty circle; inclusion is a filled-in circle.
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That way, when we have two categories, like x < 1 and 1 ≤ x, we want to be able to know which curve owns x = 1.
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And because it is less than or equal to, we would have 1 ≤ x, so it would get the dot at that point.
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And x < 1 gets to go right up to 1, but it doesn't actually get to include 1; so it uses the exclusion, the hollow, empty circle.
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Let's see an example: if we graph f(x) = x² - 1 when x < -1, 2 when -1 ≤ x ≤ 1,
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and -2x + 4, which is when 1 < x, we would see this graph right here.
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And I also just want to point out that that is what we just did in our table--our really big table,
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where we figured out all of the possibilities--that is what we just did a few slides back.
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So, x² - 1; we see this portion (I will make it in blue, actually) of the graph right here,
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because x² - 1 is a parabola; we are seeing the left portion of the parabola, because it is only
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the portion of the parabola when x is less than -1.
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We plug in those values, and we see where it gets mapped.
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However, because x is less than -1, we have exclusion right here; we are not allowed to actually use that point.
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It gets right up to it, but it can't actually touch x = -1, because it has exclusion on x < -1.
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And it is strictly less than; it is not actually allowed to equal -1.
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Similarly, we have to flip to inclusion on the rule of being 2; so it gets to include it on this.
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And then, it is just a constant from -1 up until 1; so we have this nice straight line here.
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And then, in green, -2x + 4: 1 < x; we have this thing right here.
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Now, why don't we see a dot at this junction right here?
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Well, it is because it is actually agreed on; this point shows up here, and it would show up here, if it were able to hold it.
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-2x + 4, evaluated at 2...sorry, not 2, but 1, because we are at the point 1; 1 is our changeover, right here;
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1 < x and x ≤ 1; so -2...plug in 1; plus 4; we get -2 + 4; so we get 2.
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If -2x + 4 was allowed to use x = 1, it would end up agreeing with it; so it is actually going to the exact same point.
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It is sort of a hollow for -2x + 4; but then, it immediately gets filled in for 2.
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So, ultimately, we don't see a break there.
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There is no loss, because while one end is excluding it, the other one is including it.
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And they are all in the same place, so we end up only seeing the inclusion; it is held there; they are together in that place.
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And that is one example of graphing a piecewise function.
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All right, this is a great time to bring up the idea of a continuous function.
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We are going to occasionally refer to the idea in this course, and it is going to come up a whole bunch in calculus.
00:19:06.500 --> 00:19:08.300
So, it is important to get used to it now.
00:19:08.300 --> 00:19:14.500
It is hard to formally define continuous right now; right now, we don't have enough symbolic technology.
00:19:14.500 --> 00:19:20.400
We aren't used to using symbols in the way we would need to to talk about continuous; we can't really talk about it with numbers right now.
00:19:20.400 --> 00:19:24.800
But we can understand what it means graphically; it makes great intuitive sense in pictures.
00:19:24.800 --> 00:19:30.500
All of these different things--all of these three different ways of talking about it--all mean the exact same thing.
00:19:30.500 --> 00:19:35.700
So, if a function is continuous, all of the parts of its graph are connected.
00:19:35.700 --> 00:19:40.700
Its graph could be drawn without ever having to lift your pencil off the paper.
00:19:40.700 --> 00:19:42.900
And there are no breaks in the graph.
00:19:42.900 --> 00:19:45.600
These three things all mean the exact same thing.
00:19:45.600 --> 00:19:48.600
The parts of the graph are connected; that means that there are no breaks.
00:19:48.600 --> 00:19:53.800
And if there are no breaks, then you could just put your pencil down and draw the whole thing, without ever having to lift your pencil off the paper.
00:19:53.800 --> 00:19:57.900
They are three different ways of thinking about it, but they all mean the same thing: the whole thing is connected.
00:19:57.900 --> 00:20:03.000
It is a continuous flow--this nice, connected piece of information.
00:20:03.000 --> 00:20:06.600
Great; let's look at some examples to help us understand this.
00:20:06.600 --> 00:20:11.200
For a function to be continuous, it must be any one of the below statements (we just said them; let's say them again).
00:20:11.200 --> 00:20:13.400
The parts of its graph are connected; the graph could be drawn
00:20:13.400 --> 00:20:17.900
without ever lifting your pencil or pen off the paper; and there are no breaks in the graph.
00:20:17.900 --> 00:20:21.200
So, here is one example of something being continuous.
00:20:21.200 --> 00:20:28.200
Even though it has this sort of "juke" (it suddenly changes direction there) in this corner,
00:20:28.200 --> 00:20:32.500
it is still continuous, because the graph connects in this corner.
00:20:32.500 --> 00:20:39.100
The two ends touch; you can draw it in one smooth thing without ever having to take your hand off of drawing.
00:20:39.100 --> 00:20:42.700
You could draw it in one smooth thing, without ever having to lift up.
00:20:42.700 --> 00:20:46.900
So, it is a continuous graph, at least in that viewing window--what we can see.
00:20:46.900 --> 00:20:51.100
Now, here is an example of something being not continuous.
00:20:51.100 --> 00:20:57.300
This one right here is not continuous, because we have this break; all of a sudden, we jump locations.
00:20:57.300 --> 00:21:00.900
We have an empty circle here, and we are now, all of a sudden, in a totally different place.
00:21:00.900 --> 00:21:04.900
If we were to try to draw this, we could get up to here; but to go any farther,
00:21:04.900 --> 00:21:09.200
we would have to lift our pencil up, move down, and now start down here.
00:21:09.200 --> 00:21:12.300
So, we would be in a totally different place; there is a break in the graph.
00:21:12.300 --> 00:21:18.400
The parts of the graph are not connected; it is not just one nice, connected curve; it is not a continuous function.
00:21:18.400 --> 00:21:24.100
Finally, another one that not continuous: this one is pretty close to being continuous.
00:21:24.100 --> 00:21:28.100
We can draw and draw and draw and draw and draw, but there is an empty point here.
00:21:28.100 --> 00:21:33.800
We have one point; we have this single discontinuity, this single point that has been moved off of the line.
00:21:33.800 --> 00:21:37.700
We have to move down here to draw in this single point; and then we go back to normal.
00:21:37.700 --> 00:21:42.600
So, it is really, really, really close to being continuous; but it is not perfectly continuous,
00:21:42.600 --> 00:21:49.500
because this point here has been moved down here; it is in a different place; it is not where it needs to be, to be continuous.
00:21:49.500 --> 00:21:52.300
So, a continuous function--all of the parts of its graph are connected.
00:21:52.300 --> 00:21:56.200
And that means even one point could be out, and it would break the continuity.
00:21:56.200 --> 00:21:59.700
It would break being continuous; it would no longer be connected.
00:21:59.700 --> 00:22:02.400
Great; that is an interesting function.
00:22:02.400 --> 00:22:07.200
At this point, it is probably becoming clear that you can have some kind of weird-looking functions.
00:22:07.200 --> 00:22:14.400
So far, over the course of algebra and geometry, you have seen pretty reasonable-looking things, like x², √x, x³...
00:22:14.400 --> 00:22:18.600
Even the weirdest things you have seen have been pretty reasonable--just sort of smooth curves.
00:22:18.600 --> 00:22:23.700
But we are starting to see, with piecewise functions, that things can be a little odd.
00:22:23.700 --> 00:22:28.700
Let's look at another one, a little rule--a rule that is a little more complex than x²,
00:22:28.700 --> 00:22:32.900
something that is a little more interesting than the ones we have been used to so far.
00:22:32.900 --> 00:22:40.100
So, here is an example, the step function: sometimes it is also called the greatest integer function (and it will make sense why in just a second).
00:22:40.100 --> 00:22:50.500
f(x) = double bracket x on both ends; it is just make a bracket; make another bracket; put x inside; and then close both of those brackets.
00:22:50.500 --> 00:22:54.800
The greatest integer less than or equal to x--what does that mean?
00:22:54.800 --> 00:22:57.600
The greatest integer less than or equal to x--let's try it out.
00:22:57.600 --> 00:23:06.700
What would happen if we put in 3? Well, the greatest integer that is less than or equal to 3 is 3, because 3 is an integer.
00:23:06.700 --> 00:23:13.600
And there are other integers out there, like 2; but 2 is not the greatest possible integer that is less than or equal to x,
00:23:13.600 --> 00:23:22.500
since there is -1, 0, 1, 2, 3; 3 is the greatest one that is less than or equal to 3.
00:23:22.500 --> 00:23:29.900
What if we went higher, if we said 4? Well, 4 would be greater than 3; so it is not in the running--it doesn't have a possibility.
00:23:29.900 --> 00:23:37.400
So, that would be 3; but what if we tried something that wasn't just already a straight integer, like, say, 4.7?
00:23:37.400 --> 00:23:42.900
If we plugged in 4.7, well, what is the greatest integer that is less than or equal to x?
00:23:42.900 --> 00:23:49.200
3 is a possibility; 4 is a possibility; 5 is a possibility; and it would keep going in either direction.
00:23:49.200 --> 00:24:03.700
Well, if we went from the left, let's start like this; we could say, "1--1 is a possible thing! Let's go with 1!"
00:24:03.700 --> 00:24:09.700
Oh, well, if we look at 2, it turns out that is even bigger than 1, and it is still less than 4.7; so 2 is our best option.
00:24:09.700 --> 00:24:15.800
Oh, what about 3? Well, 3 is still less than or equal to 4.7, and it is bigger than 2; so it is the best option.
00:24:15.800 --> 00:24:25.500
Oh, what about 4? 4 is bigger than 3, and 4 is less than or equal to 4.7; so it is the best option so far.
00:24:25.500 --> 00:24:37.700
What about 5? Oh, wait, 5 is greater than 4.7, so it is not in the running, because it has to be the greatest integer less than or equal to x.
00:24:37.700 --> 00:24:41.300
So, that means 5 is out of the running, and also anything larger than 5.
00:24:41.300 --> 00:24:47.200
So, everything less than 1, 2, 3...those are not going to work, because we have found 4, and it is the best so far.
00:24:47.200 --> 00:24:52.100
And everything 5 or greater isn't going to work; so that means our answer is 4.7.
00:24:52.100 --> 00:25:01.500
It is basically always rounding down; 4.7 would become 4; 3.5 would become 3; -2.5 would become -3,
00:25:01.500 --> 00:25:06.900
because we have to round down, and what is below -2.5? -3.
00:25:06.900 --> 00:25:14.100
All right, and finally, we can also sometimes call this the int(x), the integer function on x.
00:25:14.100 --> 00:25:20.600
Sometimes you will see it denoted as that; it will be written as int(x), as opposed to [[x]].
00:25:20.600 --> 00:25:24.700
It is the same idea though--this greatest integer thing, this step function where we are breaking.
00:25:24.700 --> 00:25:29.500
Now, why is it called a step function? We will look at a picture, and that will help explain it a lot.
00:25:29.500 --> 00:25:33.200
So, the graph of f(x) = [[x]] looks like this.
00:25:33.200 --> 00:25:36.600
Why does it look like this? Well, remember, at -3, where do we get placed?
00:25:36.600 --> 00:25:39.700
Well, -3 is an integer, so it just goes right here.
00:25:39.700 --> 00:25:44.900
Well, what about anything in the middle? Anything in the middle would get placed onto -3,
00:25:44.900 --> 00:25:48.100
because they would have to be rounded down to the greatest integer they are connected to.
00:25:48.100 --> 00:25:48.800
So, that is what we get there.
00:25:48.800 --> 00:25:54.700
As soon as we get to -2, though, we are going to jump up, because -2 is an integer, so it gets to be used here.
00:25:54.700 --> 00:25:59.900
And so, it is going to have the same sort of thing; anything in the middle would end up getting placed onto -2.
00:25:59.900 --> 00:26:03.500
But once we get to -1, we make it to this one; and so on and so forth.
00:26:03.500 --> 00:26:07.500
And so, we just keep stepping along and stepping along and stepping along.
00:26:07.500 --> 00:26:12.400
And every time we hit an integer, we jump up to the next height, and so on and so forth.
00:26:12.400 --> 00:26:17.000
And so, we have the greatest integer less than or equal to x, which ends up looking like a staircase,
00:26:17.000 --> 00:26:21.900
in terms of its steps, because we keep stepping up every time we hit a new integer.
00:26:21.900 --> 00:26:24.400
Cool; all right, we are ready for some examples.
00:26:24.400 --> 00:26:28.900
So, the first one just to get started: let's evaluate this function at four different points:
00:26:28.900 --> 00:26:37.300
f(x) = 3x + 10 when x < -2, 8 when x = -1, and x² - 10 when x > -1.
00:26:37.300 --> 00:26:43.600
All right, at f(-3), first what we have to do is say, "Which category do you belong to?"
00:26:43.600 --> 00:26:50.100
Well, -3 is less than -1, so it belongs to the 3x + 10 rule.
00:26:50.100 --> 00:27:04.400
So, we use 3x + 10; we plug in the -3; we have 3(-3) + 10; -9 + 10; so we have 1; f(-3) = 1.
00:27:04.400 --> 00:27:11.000
Great; what about f(-1)--what does that belong to?
00:27:11.000 --> 00:27:17.800
Well, -1 = -1, so it is using this category right here, so that means we have 8; so we have 8.
00:27:17.800 --> 00:27:22.500
There is nothing else that we have to do; it is already as simple as it can be; f(-1) = 8.
00:27:22.500 --> 00:27:24.700
And that is our answer, right there.
00:27:24.700 --> 00:27:35.200
What about f(-0.9)? This one is really close to -1; but remember, this thing was x = -1, and only happens on precisely -1.
00:27:35.200 --> 00:27:46.300
-0.9 is, in fact, slightly greater than -1; -0.9 is greater than -1, so we use the rule x² - 10.
00:27:46.300 --> 00:27:56.300
We plug in that -0.9; we have (-0.9)² - 10; -0.9 squared is 0.81; it becomes positive.
00:27:56.300 --> 00:27:59.000
Anything squared becomes positive, as long as it is a real number.
00:27:59.000 --> 00:28:07.800
0.81 - 10 becomes negative, because the 10 is bigger: -9.19.
00:28:07.800 --> 00:28:13.900
So, f(-0.9) is equal to -9.9.
00:28:13.900 --> 00:28:19.200
Finally, one more example, f(5); which category does this belong to?
00:28:19.200 --> 00:28:27.400
It pretty clearly belongs to x > -1; 5 > -1, so we use the x² - 10 rule once again; we use that process.
00:28:27.400 --> 00:28:40.000
Plug in the 5; 5² - 10 is 25 - 10, is 15; so f(5) = 15.
00:28:40.000 --> 00:28:42.300
And there we are; and that is how you evaluate a piecewise function.
00:28:42.300 --> 00:28:46.700
You see which category it belongs to; then you plug it into the appropriate rule, and you just plug it in and work,
00:28:46.700 --> 00:28:48.800
like you are doing a normal function at that point; great.
00:28:48.800 --> 00:28:54.000
The next one: all right, in this one, we will graph a piecewise function; so our function this time is:
00:28:54.000 --> 00:29:02.600
f(x) = x + 6 when x ≤ -3 and -x² - 2x + 1 when x > -3.
00:29:02.600 --> 00:29:06.500
So, first, let's make a table to help us graph this thing.
00:29:06.500 --> 00:29:15.500
x and f(x); what would be a good place to start out?
00:29:15.500 --> 00:29:22.600
Well, we have -3 showing up here and here; so that is probably going to be the midpoint, mid-"zone" in our graph.
00:29:22.600 --> 00:29:38.300
So, let's start by plugging in -3; and we will go more negative as we go up: -4, -5, -6, and we will go more positive as we go down: -2, -1, 0, 1; great.
00:29:38.300 --> 00:29:40.900
Let's try plugging in...which rule will we end up using?
00:29:40.900 --> 00:29:50.800
Well, when x is less than or equal to -3, we will end up using the things that are above -3 or equal to -3.
00:29:50.800 --> 00:30:05.400
So, this rule up here gets the x + 6 portion; and down here, when we are below the line, we get -x² - 2x + 1,
00:30:05.400 --> 00:30:11.400
because then x is greater than -3; -2 is greater than -3; 0 is greater than -3; etc.
00:30:11.400 --> 00:30:13.100
All right, so let's try doing some of these.
00:30:13.100 --> 00:30:29.900
If we plug in -3, -3 + 6 is going to equal positive 3; -4 + 6, -5 + 6, -6 + 6; what do these all come out to be?
00:30:29.900 --> 00:30:44.200
-4 + 6 gets us 2; -5 + 6 gets us 1; -6 + 6 gets us 0; so we have a pretty good idea of how to graph the x + 6, the portion of the graph where x ≤ -3.
00:30:44.200 --> 00:30:45.500
Now, what about going the other way?
00:30:45.500 --> 00:30:55.500
Well, if we plug in -2 into -x² - 2x + 1, we have -(-2)² - 2(-2) + 1.
00:30:55.500 --> 00:30:57.600
Let's plug in all of them, and then we will just do them at once.
00:30:57.600 --> 00:31:14.000
-1² - 2(-1) + 1; -0² - 2(0) + 1; -1² - 2(1) + 1; what do these all come out to be?
00:31:14.000 --> 00:31:22.500
Well, first, -2 squared becomes positive 4; so we hit that with another negative, and we have -4 right here.
00:31:22.500 --> 00:31:31.700
-2 times -2 gets us +4, + 1, so -4 + 4 gets canceled; and then + 1...we get 1.
00:31:31.700 --> 00:31:43.300
-1 squared gets us positive 1, but then, hit with another negative, we get -1; -2 times -1 gets us + 2, + 1; so -1 + 2 + 1...we get 2.
00:31:43.300 --> 00:31:48.900
0 squared gets us 0; -2 times 0 gets us 0, plus 1--we get 1.
00:31:48.900 --> 00:31:59.900
-1 squared gets us -1; -2 times 1 gets us -2, plus 1; so we get -2 here.
00:31:59.900 --> 00:32:04.300
Great; all right, at this point, we can start graphing this thing.
00:32:04.300 --> 00:32:27.000
We are graphing from -6 to 1; and our extreme y-values are...we have from 0, 1, 2, 3, so we will make it 1, 2, 3, 4, -1, -2, -3, -4, -5.
00:32:27.000 --> 00:32:36.800
That is probably enough information; I have to do down there...1, 2, 3, 4, -1, -2, -3, -4, -5.
00:32:36.800 --> 00:32:41.200
And there doesn't seem to be any reason why we shouldn't do this on a square axis.
00:32:41.200 --> 00:32:45.700
So, the tick mark length, the length of our vertical tick marks, can be the same as our horizontal tick marks.
00:32:45.700 --> 00:32:48.100
And of course, I am just doing this by hand, so it is approximate.
00:32:48.100 --> 00:32:57.100
But this isn't too bad: 1, 2, 3, 4, 5, 6, and it would keep going out that way, as well.
00:32:57.100 --> 00:33:05.000
All right, -1, -2, -3, -4, -5, and -6; positive 1, positive 2; great.
00:33:05.000 --> 00:33:08.300
So, at this point, we plot down our points, just like we are doing a normal thing.
00:33:08.300 --> 00:33:22.700
-6 goes to 0, right here; -5 goes to positive 1 here; -4 goes to positive 2 here; -3 goes to positive 3 here.
00:33:22.700 --> 00:33:25.600
And at this point, we have the line portion.
00:33:25.600 --> 00:33:32.800
Does the line keep going to the right, though? No, because it stops once it goes greater than -3.
00:33:32.800 --> 00:33:36.600
It only works, the rule only happens, when x is less than or equal to -3.
00:33:36.600 --> 00:33:44.100
But it would keep going off to the left; so it stops right here, but it does include that point, because of the "less than or equal."
00:33:44.100 --> 00:33:46.200
Now, what about the parabola part of it?
00:33:46.200 --> 00:33:59.000
Well, we plug in -2; -2 gets us 1; -1 gets us 2; 0 gets us 1; 1 gets us -2; and 2 would continue down.
00:33:59.000 --> 00:34:05.100
So, we have a pretty clear parabolic arc going on here.
00:34:05.100 --> 00:34:09.100
We are used to this; and it is going to keep going straight off forever to the right,
00:34:09.100 --> 00:34:14.200
because it is x > -3; so as long as we are continuing to go to the right, it will continue on.
00:34:14.200 --> 00:34:18.100
What happens to the left, though? We know what is happening--it is going to be in a parabolic arc.
00:34:18.100 --> 00:34:22.600
But we are not quite sure where it is going to land, because it has to stop somewhere.
00:34:22.600 --> 00:34:25.200
But we don't know what height it will stop at.
00:34:25.200 --> 00:34:33.700
We know it will stop just before -3; so it will stop at -2.9999999999999...forever, continuing forever and ever.
00:34:33.700 --> 00:34:38.400
It can't actually touch -3, but it can get infinitely close; it can get right up next to it.
00:34:38.400 --> 00:34:42.300
So, let's figure out where it would be going if it got right up next to it.
00:34:42.300 --> 00:34:53.300
What we do is: let's see what would happen if we plugged in -2.99999; now, I need even more nines, right? -2.99999999...nines forever.
00:34:53.300 --> 00:34:56.000
Now...well, not quite forever, because then it would turn into 3.
00:34:56.000 --> 00:35:03.800
But the point is -2.lots-of-nines; now, I don't know about you, but I don't want to have to plug in -2.9999999 into a calculator,
00:35:03.800 --> 00:35:06.400
because it is going to end up getting me these ugly numbers, and I will end up having decimals.
00:35:06.400 --> 00:35:13.500
And really, when you get right down to it, isn't -2.99999999 going to behave a lot like we plugged in -3?
00:35:13.500 --> 00:35:19.800
It is so close to -3 that we could probably just plug it in as if we had plugged in -3; and indeed, we can do that.
00:35:19.800 --> 00:35:25.700
We will just know that it will be an empty circle of that, because it has exclusion; it has strictly greater than.
00:35:25.700 --> 00:35:31.700
So, we will plug in -2.99999999 and 9, which still belongs to the -x² - 2x + 1,
00:35:31.700 --> 00:35:37.600
because -2.999999999 is greater than -3, if only by a little tiny bit.
00:35:37.600 --> 00:35:44.100
And it is going to behave pretty much the same as if we had plugged in -3, so we can calculate it more easily by plugging in -3.
00:35:44.100 --> 00:35:51.600
So, -(-3)² - 2(-3) + 1--what does that come out to be?
00:35:51.600 --> 00:36:02.900
-(-3)² becomes -9; -2(-3) becomes +6; plus 1, so we have -9 + 7 = -2.
00:36:02.900 --> 00:36:10.400
So, we know that this is going to go out to -2 when it gets to -3; but it is not actually going to be at -3.
00:36:10.400 --> 00:36:15.600
It is going to be hollow there, because we are excluding it--it is not actually allowed to go to that point,
00:36:15.600 --> 00:36:19.800
because the exclusion was already put on that first category on that first rule.
00:36:19.800 --> 00:36:25.200
This will curve down in a parabolic arc into the exclusion hole, and then just stop right there.
00:36:25.200 --> 00:36:30.000
It doesn't actually get to touch -3, but we can basically calculate it as if it had gotten to -3,
00:36:30.000 --> 00:36:38.200
because -2.999999999999 is so close to being just like -3, we can calculate it as if it had gotten there.
00:36:38.200 --> 00:36:42.500
But then, we just have to remember that we have to make sure that we put it in this circle here,
00:36:42.500 --> 00:36:49.600
because we are actually excluding it at -3, because x isn't equal to -3 if x > -3.
00:36:49.600 --> 00:36:54.300
Great; all those ideas that we just talked about are going to come up a lot with this.
00:36:54.300 --> 00:36:58.800
Let's go just a little bit off and pretend that we are using the real number, and then see what it is like.
00:36:58.800 --> 00:37:05.700
And we are going to do that here; so if what I do here doesn't quite make sense, look back at the explanation of that 2.999999 thing.
00:37:05.700 --> 00:37:08.700
And we will get an idea of "Oh, that is why we can do this sort of thing."
00:37:08.700 --> 00:37:19.100
So, once again, we will set this up in the same way: (x,f(x)): now what values...
00:37:19.100 --> 00:37:23.700
Clearly, -2 is kind of important; it shows up in a lot of places.
00:37:23.700 --> 00:37:31.200
What are we going to do? Well, the first thing that we are told to do in this problem is to give the domain of f; then graph it.
00:37:31.200 --> 00:37:35.800
First, let's do the domain; how do we come up with the domain?
00:37:35.800 --> 00:37:41.200
Well, remember, domain is all of the inputs that are allowed to go into a function.
00:37:41.200 --> 00:37:51.900
x² - 5 never breaks down; 3 never breaks down; -2x + 1 never breaks down; so none of the rules break down.
00:37:51.900 --> 00:38:01.600
So, none of the processes, none of the rules, break; they are always defined.
00:38:01.600 --> 00:38:06.000
However, are the categories always defined?
00:38:06.000 --> 00:38:09.900
x < -2 means we can just keep on going; we can keep on going.
00:38:09.900 --> 00:38:15.200
So, it is really negative infinity less than x; so we can go all the way down to negative infinity.
00:38:15.200 --> 00:38:18.000
What about to the right, though? Is there anything that we are not allowed to get to?
00:38:18.000 --> 00:38:22.300
Well, negative infinity up to -2; and then -2 is here at equal; and then -2 is less than...
00:38:22.300 --> 00:38:29.000
so we have covered all of our bases, from negative infinity up to -2; and keep going, up until 1.
00:38:29.000 --> 00:38:32.900
Are there any rules for what happens if x is greater than 1?
00:38:32.900 --> 00:38:43.900
No, we don't have any rules; we have x < -2, x = -2, and -2 < x ≤ 1; but we don't have any rules for when x > 1.
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So, no rules for x > 1 means that f doesn't tell us what to do if we are plugging something in.
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The f fails to tell us what to do to this input if we plug in something that is greater than 1.
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If we plug in, say, 500, we look at this, and we say, "Oh, this doesn't belong to any categories."
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So, f is undefined at 500; it doesn't work; it is not in the domain.
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So, the domain fails to contain everything in x > 1; so that means our domain is not going to be x > 1.
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That is the things of failure, because we don't have rules; the domain is everything from negative infinity
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(we use a parenthesis for negative infinity, and infinity), and we go up until 1; and we include the 1,
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because we have less than or equal to, but we can't go past it; we have no more rules to go up past it.
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So, f has a domain from negative infinity up until 1, including 1.
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Great; now, let's build up that table.
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-2 seems like a good place to make our middle; and if we are above -2, we will use which rule?
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We use the x² - 5 rule.
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Sorry, by "above," I meant to say more negative than -2.
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And if we are below on this table, which is to say more positive, closer to 0, we are going to use -2x + 1.
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So, -2...we will have -3 and -4, -1, 0, 1...but just like we did in the last thing, it will be useful to know
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where it is going if it had been allowed to get to -2.
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So, for the above part, we will say -2.0001; and -1.9999; these things are because -.1999999 is greater than -2,
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and -2.000001 is less than -2, but they are going to behave effectively as if we had plugged in -2.
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So, when we are actually figuring out the numbers, we can pretend as if we had plugged in -2, just to make it easier on us to do the calculation.
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All right, the first one, -2: what are our f(x)'s?
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-2's rule just says to give out 3; it doesn't matter what your input is, even though we have to use the category of -2.
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So, it automatically gives out 3 at -2.
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What about -2.00001, which would use the x² - 5 rule? Well, that is about the same thing as plugging in -2.
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We have (-2)² - 5; keep going--let's just keep going up to get them all written out.
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(-3)² - 5, and (-4)² - 5--what do those all equal?
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Well, (-2)² becomes 4; 4 - 5 is -1; (-3)² is 9; 9 - 5 is not -4, but +4; 9 isn't bigger than -5.
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And (-4)² is +16; 16 - 5 is 11; OK.
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What about the other way, if we go to the -2x + 1 rule?
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Well, if we had plugged in -2, we would get -2 times -2; we aren't literally plugging it in.
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We are just saying, "What if we had gone all the way up to it? Let's see what would have happened,"
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even though ultimately we will have to exclude it, because we have these strictly less than and strictly greater than signs.
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So, -2 times -2 plus 1; -2 times -1 plus 1; -2 times 0 plus 1; 1 times 0...sorry, not 1; sorry about that...
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-2 times positive 1 (I got that confused with the one above it); -2 times 1 plus 1; what do those all equal?
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The thing that is effectively going to be like -2...-2 times -2 is positive 4; 4 plus 1 is 5.
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-2 times -1 is positive 2, plus 1 is 3; -2 times 0 is 0, plus 1 is 1; -2 times 1 is -2, plus 1 is -1.
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Great; so now we are in a position to be able to graph it.
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Our extremes...vertically we can get up to really high things when we are in the x² - 5; so we won't worry about the 11 part.
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But we are going between extremes of 5, maybe a little lower; so we will graph this...
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We never get to very low values, it seems; so we will put our corner down here.
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And we also never get past 1; remember, our domain is only -∞ up until 1.
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So, we also don't have to have a whole lot of stuff on the right.
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So, we have positive 1 here, positive 2 here, positive 1, positive 2, positive 3, positive 4, positive 5, positive 6; -1, -2, -1, -2, -3, -4, -5.
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Great, and that is plenty of room, because we only get up to -4; and we know that x² - 5 is going to blow out.
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So, 1, 2, 3, 4, 5, 6, -1, -2, 1, 2, 3, -1, -2, -3, -4, -5; making tables...making axes.
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So, let's plug in some things and see what happens.
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We plug in -4, and it goes out to 11; so we can't even see it; it is so high up.
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-3 gets to 4, though; we can definitely plot that; so -3 goes to 4, (-3,4).
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-2, if it had a -2...it doesn't actually have it, but we know that -2.0001 would practically be going to -1.
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So, we are putting an exclusion hole down here, just below the -2.
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At -2, though, we actually end up being at 3; so we have this single point right there.
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If we had -2 for the -2x + 1 process, we would be at 5; but we can't actually go there, so once again, we have an exclusionary hole there.
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-1 is at 3; 0 is at 1; 1 is at -1; and it stops right there, because we stop at 1.
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We can't go any higher than positive 1; our domain caps out there.
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So, our straight line is just a straight line, up until where it stops at that exclusionary hole.
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We have this point in the middle, the 3 point, but it is just x = -2; and then we have a parabola curving up.
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It is already moving pretty fast by this point, so it is not going to be a nice, smooth parabola like this.
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It is already moving fast up, because it is pretty far up.
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It manages to jump from -1 to 4, and then from 4 to 11; so it is not the bottom part of the parabola.
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It is already in process, in a way; so curve this parabola up, and it just zooms way, way off really quickly.
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All right, that is basically what we are seeing here for our graph.
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We have a parabola on the left-hand side, which drops to the single point--there is just a single point in the middle.
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And then, we switch to -2x + 1, which goes to 1, and then stops at 1,
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because the category just stops at 1; so it stops right here, and we don't have anything farther to the right.
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There is nothing further off to the right, because the categories don't include anything further right.
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All right, the final example: A certain phone company charges $20 for using its service, along with 10 cents for each minute under 200 minutes.
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After 200 minutes, they charge 5 cents for each additional minute.
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Let's give a piecewise function, p(t), price in terms of t, that will describe the price in terms of t, the minutes spoken.
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So, t is the minutes spoken.
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It is pretty easy for us to figure out what the first part is.
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The first portion: when we are under 200 minutes, which is to say when t ≤ 200, the price of t is not too hard for that.
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p(t) =...well, a $20 flat rate--they charge us $20, and then they charge us 10 cents for each additional minute.
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So, $20 plus that additional 10 cents...how many minutes did we have? We had t.
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So, 0.1t--that is what it is; let's do a really quick test--let's say if we had talked 100 minutes.
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Then 100 minutes times 10 cents would be $10, so we would have a $30 total,
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which, if we plug that into our new function that we just made, p(t), p(100) would be 20 + .1(100), which would come out to be 30.
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Great, so the first part of it checks out.
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What about the second portion, though?--that is where things start to get a little complicated.
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So, in the second portion, when we are over 200, which is to say t > 200...
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and actually can be greater than or equal, because we know that they are going to have to agree;
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there is not going to be a sudden jump there; and we will talk about that more later.
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It is a way of checking this function, actually.
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We know that it is going to be 5 cents for each additional minute.
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Our first thought might be, "Oh, great, easy; it is going to be 0.05t."
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Not true--this is not going to be the case.
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Why not? Because it is for each additional minute, over 200; so after 200, you get charged at 5 cents per minute.
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Before that, you still get charged at the 10 cents; so how many minutes over 200?
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Well, that is not too hard; we know that we have t minutes total.
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We know that we are already over 200, so it is going to be the number of minutes we have talked, minus 200.
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So, t - 200 is the number of minutes we talked; so it is 0.05 cents, times t - 200.
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Now, that is the amount of additional money that will be on top of some lump.
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How much is it to even make it to 200 minutes in the first place?
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Well, 200 minutes in the first place: let's see what it is from our first one.
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p(200) would be equal to 20 + 0.1(200); just move the decimal place over one, so it becomes 20; so 20 + 20 is 40.
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So, it costs $40 to get up to 200 minutes; so it costs $40 at 200, and then it is plus the additional amount per minute.
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So, for the second portion, our function is going to be p(t) = $40, the lump sum that we have to pay at first
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to have even made it to the 200-minute mark, plus 5 cents for the number of minutes over 200 minutes.
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So, our function has been broken into 2 pieces; so we have a piecewise function here, p(t) = 20 + 0.1t when t ≤ 200,
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and 40 + 0.05 times the minutes over 200 when t ≥ 200.
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Now, we know that the two have to agree; otherwise, people would make sure to make that jump or not make that jump,
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because otherwise with the sudden change, or the switch--it wouldn't make sense for the phone company
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to have it suddenly leap more on your bill or cut off a portion of your bill if you were to hit the 200 mark.
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It is going to just continue in a continuous function, we would expect.
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So, we can check this; and we can check and make sure that, indeed, p(200) = 20 + 0.1(200).
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We already did this before; it was $40; and let's check and make sure that the second portion, p(200), would agree.
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40 + 0.05 at the minutes over 200, so that is 200; 200 - 200 is just 0, so that cancels out the 0.05; so we get 40.
00:51:07.700 --> 00:51:14.700
So, those two things check, and our function, price in respect to time, makes perfect sense.
00:51:14.700 --> 00:51:20.500
So, p(t) = 20 + 10 cents per minutes when minutes are less than or equal to 200,
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or 40 + 5 cents per minutes over 200, when the number of minutes is greater than or equal to 200.
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Great; I hope piecewise functions are making a lot more sense now.
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Remember: it is an idea about putting into the category, then applying the rule based on the category.
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That is the prime, the major idea in piecewise functions; if you can hang onto that, you will be able to make sense of them.
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All right, we will see you at Educator.com later--goodbye!