WEBVTT mathematics/algebra-2/eaton
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Welcome to Educator.com.
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In today's session, we are going to talk about several special functions.
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The first one we are going to discuss is the **step function**; and the step function makes a unique-looking graph.
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It is a function that is constant for different intervals of real numbers.
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And the result is a graph that is a series of horizontal line segments, so they look like steps; and that is where the name "step function" comes from.
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The best way to understand this is through an example.
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So, for example, if apples were sold at a price of a dollar per pound, and the price is such that
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you are charged $1 for each pound, or any part of a pound--in other words, they round up in order to determine the price;
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if you had a pound and a half of apples, they are going to charge you $1 for the pound, and then another $1 for the half pound.
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So, they charged you a dollar for a pound, and then any part of a pound is considered a full pound in terms of pricing.
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So, if x is the number of pounds, and y is the cost, then let's see what kind of values we get and what the graph looks like.
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OK, if I have .8 pounds, I am going to get charged $1: they are going to round up.
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If I have a full pound, I am going to get charged $1; if I have a little bit over a pound--I have 1.2 pounds--
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I am going to get charged $1 for the first pound, and then that .2 is going to be another dollar, so $2.
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1.4 pounds--again, $2, and on up...2 pounds--a dollar for the first pound and a dollar for the second pound.
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2.5--$2, and then the .5 is another $1, so that bumps me up to $3; until I hit 3 pounds, it is $3, also.
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3.8: $1 for the first pound, $1 for the second pound, $1 for the third pound, and another $1 for that .8, so $4.
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So, you can see that the function is constant for different intervals.
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So, for this first interval, from a little bit above 0, all the way to 1, the y-value is constant; it is $1.
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For the next interval, which is just above 1, all the way to 2, including 2, it is going to be $2.
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Once I get above 2, up to and including 3, it is $3; and so on.
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So, this is constant for different intervals of real numbers.
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Looking at what the graph looks like: any part of a pound up to and including a dollar for that first pound is a dollar.
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Now, 0 pounds of apples are going to be $0; so I am not going to include 0.
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But even .1...the slightest part of a pound is going to be $1, so I put an open circle to indicate that 0 is not included;
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but just above that is, all the way up to and including 1; since 1 is included (1 is also a dollar), I am going to put that as a closed circle.
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So here, I have pounds; and here, I have the cost on the y-axis in dollars.
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Now, once I get just above 1 (say 1.1 pounds), they are going to charge me $2--not including 1, but just above it--open circle.
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All the way up to 2...at 2 pounds, I will also be charged $2.
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Once I hit just above 2, I am going to be charged $3, all the way to 3 and including 3.
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And on...so, you can see how this looks like a series of steps, and how this is a result of the fact that the function is constant for different intervals of x.
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This function is the same for this entire interval; then it is the same for the second interval; and on.
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So, this is a step function.
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A second type of function that you will be working with is an **absolute value function**.
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And these functions have special properties: looking first at f(x) = |x|, just the simplest case, here f(x) is just going to be the absolute value of x.
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When x is 0, f(x) is also 0; when x is 1, the absolute value is 1; and so on for positive numbers.
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Now, let's look at negative numbers: for -1, the absolute value is 1; -2--the absolute value of x is 2; and on.
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The result is a certain shape of graph: when x is 0, f(x) is 0; x is 1, f(x) is 1; x is 2, f(x) is 2; and on.
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Now, for negative numbers: -1, f(x) is 1; -2, f(x) is 2; -3, f(x) is 3; and it is going to continue on like that.
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So, absolute value graphs are v-shaped; so we are going to end up with a v-shaped graph.
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Depending on the function, the graph can be shifted up, or it can be shifted to the right or to the left; and let's see how that could happen.
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Let's now let f(x) equal (here f(x) equals |x|)...let's say f(x) equals the absolute value of x, plus 1.
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OK, so we are given x, and the absolute value of x is 0; we are adding 1 to them, so this is going to become 1.
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The absolute value of x is 1; 1 + 1 is 2; 3; 4; the absolute value of -1 is 1; 1 + 1 is 2.
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The absolute value of -2 is 2; add one to that--it is 3; and add 1 to 3 to get 4.
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OK, so it is the same as this, except increased by 1: each value of the function has been increased by 1.
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So now, let's see what my graph is going to look like.
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Right here, I have the graph for f(x) = |x|; now, I am going to look at this graph.
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When x is 0, f(x) is 1; when x is 1, f(x) is 2; when x is 2, f(x) is 3; so you can see what is happening.
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And then here, I have x is 3, f(x) is 4; negative values--when x is -1, f(x) is 2; when x is -2, f(x) is 3; when x is -3, f(x) is 4.
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OK, I am drawing the line through this: this is the graph of f(x) = |x| + 1.
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So you see that this graph, the v-shaped graph, is simply shifted up by 1.
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And again, you can also shift this from side to side; and we will see an example of that later on.
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So, in the absolute value function, it is very important to find both negative and positive values of the function.
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So, assign x 0; assign it some positive values; and it is very important to find what f(x) will be when x is negative,
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because if I didn't--if I picked only positive values--I would end up with half of a graph.
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So, to get the entire v-shape, choose negative and positive values for x.
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The third special function that we are going to discuss is called a **piecewise function**.
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And a piecewise function is a function that is described using two or more different expressions.
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The result is a graph that consists of two or more pieces.
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Just starting out with one that consists of two pieces: the notation is usually like this--one large brace on the left:
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f(x) equals x + 2 for values of x that are less than 3; and f(x) equals 2x - 3 for values of x that are greater than or equal to 3.
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So you see that, for different intervals of the domain, the function is defined differently.
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So, it is a function that is described using two or more different expressions.
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So, for the part of the domain where x is less than 3, this is the function.
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For the interval of the domain that is greater than or equal to 3, this is the function that I am going to use.
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Let's see what happens: let's first use this part of the function where f(x), or y, equals x + 2.
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And for the domain, remember that x is going to be less than 3.
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So, I will go ahead and start out with 2: when x is 2, f(x) (or y) is 4.
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When x is 1 (I have to remain at x-values less than 3), f(x) is 3; when x is 0, f(x) is 2.
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Just picking a negative number: when x is -4, f(x) is -2.
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I am going to graph that here; OK, when x is 2, f(x), or y, is 4; x is 1, y is 3; x is 0, y is 2; x is -4, y is -2.
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OK, now, this is for values of x that are less than 3; 3 is about here.
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Therefore, anything just below 3, but not including 3, is going to be part of this graph.
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So, we are going to use an open circle here to indicate that 3 is not going to be included as part of this function--the domain of this function.
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So, it is going to begin at just below 3 and continue on indefinitely; that is the first piece of the graph.
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The second piece of the graph is for x such that x is greater than or equal to 3; and here, y is going to be 2x - 3.
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So, it is greater than or equal to 3, so I am first going to let x equal 3.
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3 times 2 is 6, minus 3 is 3; getting larger--when x is 4, it is 2 times 4 is 8, minus 3 is 5; when x is 5, 5 times 2 is 10, minus 3 is 7.
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Now, this does include 3--this section of the graph--this piece; so, when x is 3, y is 3, right here.
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When x is 4, y is 5, right here; when x is 5 (let me shift that over just a bit), y is 7; OK.
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Now, if you look, this actually did end up including all possible values of x (all real numbers),
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because when x is less than 3, I use this function; and then, as soon as x becomes 3 or greater, I shift to this other function.
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So, you can see how there are two pieces to the graph; and you actually can have situations where there are more than 2.
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You could be given, say, f(x) is 4x + 3, 2x + 7, and x - 1, and then given limits on the domain for each of those.
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So, there are at least two pieces; however, there can be more.
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In Example 1, we have a greatest integer function.
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Before we start working in this, let's just review what we mean by the greatest integer function.
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So, when you see this notation with the brackets, let's say that you have a number in here, such as 4.7.
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What this is saying is that this value is equal to the greatest integer that is less than or equal to 4.7; so that is 4.
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It is the greatest integer less than or equal to whatever is in here; if it was 2.8, it would be 2.
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Be careful with negative numbers: let's say I have -3.2--the temptation is to say, "Oh, that is equal to -3";
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but if you look at it on the number line, -3.2 is right about here; OK, so if I have -3.2,
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and I am trying to find the greatest integer that is less than or equal to 3.2, it is going to have to be something over here--smaller.
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So, it is actually going to be -4; so just be careful when you are working with negative numbers.
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Whatever is inside here--whatever that value is--the function is equal to the greatest integer that is less than or equal to this value in here.
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Understanding that, you can then find the graph; so let's find a bunch of points for this,
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so we make sure we know what is going to happen with various situations.
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When x is 0, this inside here is going to be 2; the greatest integer less than or equal to 2 is 2.
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When x is .6, then you are going to get 2.6 in here; the greatest integer less than or equal to 2.6 is 2.
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.8--I get 2.8; again, I round down to 2.
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All right, so when I hit 1, 1 plus 2 is 3, and the greatest integer less than or equal to 3 is 3.
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Slightly above 1: that is going to give me 1.2 + 2 is 3.2; the greatest integer less than or equal to 3.2 is also 3.
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OK, so you can get the idea of what this is going to look like.
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And that continues on; and then, when we hit 2, 2 + 2 is 4; the greatest integer is going to be 4.
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For negative numbers: let's take -.5: -.5 and 2 is 1.5; the greatest integer less than or equal to 1.5 is going to be 1.
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Now, notice: I have a negative number for x, but this did not come out to be a negative number; so that is different from the case I was discussing there.
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Let's go a little bit bigger--let's say -3: -3 and 2 is -1, and that is going to be -1.
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Let's say I take -3.5: -3.5 and 2 is going to equal -1.5: again, just thinking about that to make sure you have it straight,
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I have -1.5; so I have 0; I have -1; I have -1.5; I have -2; the greatest integer less than or equal to this is actually -2.
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OK, now plotting this out: when x is 0, f(x) is 2; when x is slightly above 0 (it's .6), f(x) is 2; .8--it is 2, all the way up until I hit 1.
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At 1, f(x) becomes 3; therefore, 1 is not included in this interval.
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So, you can already see that this is going to be a step function, because we have intervals.
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For different intervals of the domain, we have that same value for the range.
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All right, for values between 1 and 2, f(x) will be 3; once we hit 2, I have to do an open circle, because at 2, the value for f(x) jumps up to 4.
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OK, so you can see what this is going to look like; and that pattern is just going to continue.
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Let's look over here at negative numbers: when x is slightly less than 0, then you are going to end up with an f(x) that is 1.
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So, for values slightly less than 0, but not including 0, this is what you are going to end up with.
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OK, looking, say, when x is -3: when x is -3, f(x) will be -1.
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But when we go slightly more negative than that, when x is -3.5, f(x) is going to be -2; it is going to be down here.
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So, the steps on this side are going to have the open circle on the right.
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And I am going to jump down, and it is not going to include -2, because -2 and 2 is 0;
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so -2 is going to be right here for the x-value, and the f(x) will be 0.
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But as soon as I get to a little bit bigger than -2, the greatest integer is going to be down here.
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OK, and so, we continue on like that with the steps; and you can see how this is a step function.
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You just have to be very careful and pick multiple points until you can see the pattern
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where for a certain interval of the domain, the range is a particular value.
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OK, so that was a step function, and it involved the greatest integer function.
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Example 2: now we are working with absolute value.
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g(x) equals the absolute value of x, minus 3.
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And we already know that the shape of this graph is going to be in a v.
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But we don't know exactly where that v is going to land, so let's plot it out.
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When x is 0, the absolute value of x is 0; minus 3--that gives me -3.
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When x is 1, the absolute value is 1; minus 3...g(x) is -2.
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When x is 2, the absolute value is 2; minus 3 is going to give me -1.
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Now, let's pick some negative numbers for x, because that is really important to do with an absolute value graph.
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When x is -1, the absolute value is 1, minus 3 gives me -2; you can already see that my v shape is going to occur.
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When x is -2, the absolute value is 2; minus 3 is -1.
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The absolute value of -3 is 3; minus 3 is 0; so this is enough to go ahead and plot.
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x is 0; g(x) is -3; x is 1, g(x) is -2; x is 2, g(x) is -1; over here with the negative values,
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when x is -1, g(x) is -2; when x is -2, g(x) is -3; when x is -3, g(x) is 0.
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So, you can see that I have a v-shaped graph, and compared with my graph that would look like this,
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that would have the v starting right here, it has actually shifted down by 3; that is an absolute value function.
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Here you can see that you are given a piecewise function, because there are two different pieces.
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And this could also be written in this notation.
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There are two different sections to the graph; and we see that the function is defined differently for different intervals of the domain.
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Starting with if x is greater than 2 (this is going to be for x-values where x is greater than 2): f(x) is going to be x + 1.
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When x is 3, f(x) is 4; when x is 4, f(x) is 5; when x is 5, f(x) is 6; OK.
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When x is 3, f(x) is 4; when x is 4, f(x) is 5; and it is going to go on up.
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And that is going to go all the way, until just greater than 2.
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2 is not going to be included on this graph, because it is a strict inequality; so I am going to use an open circle, and this is going to continue on.
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Now, for x less than or equal to 2, I have a different situation: I am looking at f(x) is -2x.
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OK, so when x is 2, 2 time -2 is -4; when x is 1, 1 times -2 is -2; when x is 0, f(x) is 0; when x is -2, that is -2 times -2, which is positive 4.
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So, starting with x is 2: when x is 2, f(x) is -4; and that is including the 2.
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When x is 1 (these are values less than or equal to 2, so I am getting smaller), f(x) is -2.
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x is 0; f(x) is also 0; when x is -2, f(x) is up here at 4; OK, so I have a steep line going right through here.
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So, you can see: this is a piecewise function consisting of two pieces; and here, one picks up where the other leaves off.
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For values greater than 2, this is my graph; for values of x less than or equal to 2, this is my graph; so this is a piecewise function.
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OK, this time, in Example 4, we have both greatest integer and absolute value in this function.
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Recall that, for the greatest integer function, what that is saying is that whatever is inside this bracket--let's say it's 1.2--
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it is asking for the greatest integer less than or equal to 1.2; in that case, this would be 1.
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Or if I had 4.8, it would be 4.
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For negative numbers, like -3.2, the greatest integer less than or equal to -3.2 is -4.
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OK, now, since this is a bit complicated, it is helpful just to take it in stages.
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So, I am going to look first at what the greatest integer of x is; and then, I am going to look for the absolute value of what that is.
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If x is .2, the greatest integer less than or equal to .2 is 0; the absolute value of 0 is 0; so this is the function that we are looking for.
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And the same would hold true of .5: round down to 0; the absolute value is 0.
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When we hit 1, the greatest integer less than or equal to 1 is 1, and the absolute value of that is 1.
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1.2: again, we are going to go down to the greatest integer that is less than or equal to 1.2, which is 1; and the absolute value is 1.
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The same for 1.8, and all the way up until 2; once we hit 2, the greatest integer less than or equal to 2 is 2; the absolute value is 2.
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So, that is working with positive numbers, greatest integer, and the absolute value; it is the same; OK.
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So, let's go to negative: for -.4, the greatest integer that is less than or equal to -.4...I am looking, and I have 0, and 1,
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and -1, and -.4 is about here; so I am going to go down to -1; the absolute value of that is 1.
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Here you can see that the greatest integer is not the same as the absolute value.
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Or for -1, the greatest integer less than or equal to -1 is -1; the absolute value is 1.
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-1.8: the greatest integer that is less than or equal to -1.8...I am going to go down to -2; and the absolute value is 2.
00:28:06.200 --> 00:28:14.200
For -2, the greatest integer less than or equal to -2 is -2; the absolute value is 2.
00:28:14.200 --> 00:28:24.200
So, you see that there are intervals here--intervals of the domain end up with the same value for the function.
00:28:24.200 --> 00:28:28.900
So, I am going to have a step function.
00:28:28.900 --> 00:28:35.000
But remember that absolute value graphs are v-shaped, so I am going to end up with a v-shaped step function.
00:28:35.000 --> 00:28:46.300
Let's plot these out: for 0, the greatest integer of 0 would be 0, and then the absolute value would be 0.
00:28:46.300 --> 00:28:56.100
So, with 0, we are going to include it; and for all values up to but not including 1, the function is going to equal 0.
00:28:56.100 --> 00:28:59.700
Once we get to 1, I have an open circle, because it is not included.
00:28:59.700 --> 00:29:06.100
When x is 1, f(x) is 1; so I am going to jump up here.
00:29:06.100 --> 00:29:14.500
All the values between 1 and 2, but not including 2, will have an f(x), or a y-value, that is 1.
00:29:14.500 --> 00:29:31.000
As soon as I hit 2, open circle: I am going to jump up, and once I hit 2, f(x) is 2, all the way up to, but not including, 3.
00:29:31.000 --> 00:29:37.800
And it is going to go on that way: and you see now, we have the step function, and it is v-shaped like absolute value.
00:29:37.800 --> 00:29:41.900
Let's look over here on the negative side of things.
00:29:41.900 --> 00:29:59.200
For -.4, somewhere in here, it is going to equal 1; -1 is also equal to 1; so here, on the left side, I have a closed circle, and an open circle on the right.
00:29:59.200 --> 00:30:02.300
It is the opposite of what I had over here.
00:30:02.300 --> 00:30:15.500
When I get to less than -1, my value for f(x) is going to jump up to 2; this is a closed circle;
00:30:15.500 --> 00:30:21.700
I get slightly less than, but not including, -1; it is going to jump up to 2.
00:30:21.700 --> 00:30:41.400
-2: my value is also 2, and everything in between; and then, when I get to just slightly more negative than -2, like -2.1, it is going to jump up to 3.
00:30:41.400 --> 00:30:47.100
You can see how this is v-shaped, and it is a step function.
00:30:47.100 --> 00:30:52.700
The step function comes from it being the greatest integer function; the v shape comes from that absolute value.
00:30:52.700 --> 00:31:05.000
And you also just had to be careful how you are doing the open and the closed circles; OK.