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Piecewise Functions
 Up until now, all the functions we've seen used a single "rule" over their entire domain. No matter what goes into the function (as long as it's in the domain), the same process happens to the input. In this lesson, though, we'll examine functions where the process changes depending on what goes into the function. This is the idea behind piecewise functions.
 As a nonmath example, we could imagine cooking potatoes in different ways depending on the size of the potato. If it's small, you boil and mash the potato. But if it's over a certain size, you cut it up and turn it into fries. In both cases, something happens to the potato. But the transformation is different depending on the type.
 We use the following notation for piecewise functions:
That is, given some input, we first check which category it belongs to, then use the corresponding transformation. [Check out the video for some concrete examples of this.]f(x) = ⎧
⎪
⎪
⎨
⎪
⎪
⎩Transformation Rule 1, x is in Category 1 Transformation Rule 2, x is in Category 2 Transformation Rule 3, x is in Category 3 : :  We graph piecewise functions the same way as other functions: a series of points ( x, f(x) ). The difference is that the rule determining where x maps to can change depending on which x we're looking at. Often it looks like the graph "changes" at switchovers between rules: when x switches from one category to another, the shape/location of the curve can change.
 An important graphical note is that we show inclusion with a solid circle. We show exclusion with an empty circle. That way, when we have two categories like x < 1 and 1 ≤ x, we can see which curve in the graph "owns" x=1.
 This is a great time to bring up the idea of a continuous function. We'll occasionally refer to this idea in this course, and it will come up often in Calculus. It's hard to formally define continuous with symbols and numbers right now, but we can understand what it means graphically. All the below mean the same thing:
 All the parts of its graph are connected.
 Its graph could be drawn without ever having to lift your pencil off the paper.
 There are no "breaks" in the graph.
Piecewise Functions

 This function is a piecewise function. Depending on the input that goes in, different transformations are applied. To figure out the transformation that should be applied, look at the category attached to each transformation rule. For each of the inputs, check which category the input falls in, then use the associated transformation.
 For f(−1), we have −1 ≤ 2, so it is in the first category: x ≤ 2. Thus,
f(−1) = 3 (−1) − 3.  For f(2), we have 2 ≤ 2, so it is in the first category: x ≤ 2. Thus,
f(2) = 3 (2) − 3.  For f(4), we have 4 > 2, so it is in the second category: x > 2. Thus,
f(4) = (4)^{2} +7.

 This function is a piecewise function. Depending on the input that goes in, different transformations are applied. To figure out the transformation that should be applied, look at the category attached to each transformation rule. For each of the inputs, check which category the input falls in, then use the associated transformation.
 For g(−6), we have −6 ≤ −1, so it is in the first category: t ≤ −1. Thus,
g(−6) =
√3 − (−6)+ 2.  For g(3), we have −1 < 3 ≤ 3, so it is in the second category: −1 < t ≤ 3. Thus,
g(3) = − 20 (3)+2.  For g(8), we have 8 > 3, so it is in the third category: t > 3. Thus,
[Notice that the third transformation rule is just a constant function. As long as t > 3, the function will always output 47.]g(8) = 47.

 This function is a piecewise function. Depending on the input that goes in, different transformations are applied. To figure out the transformation that should be applied, look at the category attached to each transformation rule. As you draw the function's graph, notice that each transformation rule will only apply as long as the x comes from its associated category.
 For x < −1, draw the graph of y=−x−3. Remember that this only applies for x < −1, so the graph of y=−x−3 will only exist over that portion of the xaxis. Once we go past x=−1, the graph will change to something else.
 If you're not sure how to draw the graph of y = −x−3, create a table of values and plot points until you see how to graph it. But remember, it only graphs for x < −1.
 For x ≥ −1, draw the graph of y = x^{2} −4. Remember that this only applies for x ≥ −1, so the graph of y=x^{2}−4 will only exist over that portion of the xaxis. If we look to the left of x=−1, the graph will change to something else.
 If you're not sure how to draw the graph of y = x^{2}−4, create a table of values and plot points until you see how to graph it. But remember, it only graphs for x ≥ −1.
 The only thing left to do is to designate what happens at any "switchover" locations, where it changes between transformation rules. We show inclusion with a solid circle and exclusion with an empty circle. Since x < −1 does not include x=−1, its associated graph (y = −x−3) has an empty circle there. For the other portion, because x ≥ −1 does include x=−1, its associated graph (y = x^{2} −4) has a solid circle there.

 This function is a piecewise function. Depending on the input that goes in, different transformations are applied. To figure out the transformation that should be applied, look at the category attached to each transformation rule. As you draw the function's graph, notice that each transformation rule will only apply as long as the x comes from its associated category.
 For x < −3, draw the graph of y=x+5. Remember that this only applies for x < −3, so the graph of y=x+5 will only exist over that portion of the xaxis. Once we go past x=−3, the graph will change to something else.
 If you're not sure how to draw the graph of y = x+5, create a table of values and plot points until you see how to graph it. But remember, it only graphs for x < −3.
 For x=−3, the function has a value of g(−3)=7, so plot a point at (−3, 7).
 For x >−3, draw the graph of y = √{x+3}. Remember that this only applies for x >−3, so the graph of y=√{x+3} will only exist over that portion of the xaxis. If we look to the left of x=−3, the graph will change to something else.
 If you're not sure how to draw the graph of y = √{x+3}, create a table of values and plot points until you see how to graph it. But remember, it only graphs for x > −3.
 The only thing left to do is to designate what happens at any "switchover" locations, where it changes between transformation rules. We show inclusion with a solid circle and exclusion with an empty circle. Since x < −3 does not include x=−3, its associated graph (y = x+5) has an empty circle there. For x=−3, it clearly includes that xlocation, so it has a solid circle. Because x > −3 does not include x=−3, its associated graph (y = √{x+3}) has an empty circle there.

 [ [t ] ] is the step function: it produces the greatest integer less than or equal to t.
 For h(x), we have more than just x inside the brackets, so we round down based on the entirety of what is inside the brackets. For example:
h(3) = 3· ⎡
⎣[0.5 (3) ] ⎤
⎦= 3 ⎡
⎣[1.5 ] ⎤
⎦= 3 ·1 = 3  Using the above understanding, we can create a table of values that will allow us to graph the function. As you create the table of values, notice that the graph "steps up" at intervals of length 2.
 For each of the steps, we need to appropriately show what happens at the "switchover" location with either an empty circle (exclusion) or a solid circle (inclusion). The left side of each chunk will get the solid circle, because the left side is included each time. For example h(2) = 3, but for any value less than x=2, the function will output less than 3. Thus, x=2 is included on the line chunk at height 3. By this same logic, the right side of each chunk will get an empty circle, because it must be excluded each time.

 We already know what each of the transformation rules are, we just need to figure out where the "switchovers" occur. Look at the graph to figure out what portion of the graph "belongs" to each of the functions.
 The first transformation rule switches over at x=−1. Since y=2x+4 does not have the end point at x=−1 (which we see because of the empty circle), the category for the first transformation rule is x <−1.
 The second transformation rule starts at x=−1. Since y = −x^{3}+1 has the end point at x=−1 (which we see because of the solid circle), we have −1 ≤ x ???? so far for the second category.
 By looking at the graph, we see that it switches from y = −x^{3}+1 to y = −7 at the horizontal location x = 2. However, there is no jump in the graph there. Both y = −x^{3}+1 and y=−7 output the same value for x=2, so they "agree" on the location. Since either transformation rule can be considered to "own" the point, there are two possibilities. Either the second category contains it (−1 ≤ x ≤ 2), or the third category contains it (x ≥ 2). Either of these two possibilities is correct, but we can't have both simultaneously, because it doesn't make sense to allow both categories to contain the location.


 Graph each portion of f(x) over the appropriate xlocations.
 Don't forget to put a solid/empty circle at the end of each portion depending on if it is included/excluded.
 A function is continuous when all the parts of its graph are connected and it can be drawn without any "breaks".
No, f(x) is not continuous because it has a break at x=2.

 Graph each portion of g(x) over the appropriate xlocations.
 If there is no "break" or "jump" in the graph when it switches transformation rules, then there is no need to put a solid/empty circle, because the value of the function is clearly defined at that xlocation.
 A function is continuous when all the parts of its graph are connected and it can be drawn without any "breaks".
Yes, g(x) is continuous because it has no breaks anywhere.
Give a piecewise function p(b) that describes the price of purchasing b books.
 For the first twenty books, the price is simply the number of books (b) multiplied by the cost of each book ($9).
 After 20 books though, things change. At first, we might be tempted to simply multiply the number of books (b) by the new price ($5), but notice that will not take into account the fact that the first 20 books cost $9.
 We have to account for the change in cost by still charging $9 for the first 20 books, then change the price only for the remaining books.
 We have b−20 remaining books which will get the discounted price. Thus, if we buy more than 20 books, the cost comes out to
9 ·20 + 5(b−20)  To create the piecewise function, notice that the first pricing scheme occurs only when 0 ≤ b ≤ 20, while the second pricing scheme only happens for b > 20.

Give a piecewise function p(t) that describes the price of being a member at Hooke's gym for t months.
 For the first year (12 months), the price is simply the number of months (t) multiplied by the cost of each month ($10).
 After 12 months though, things change. At first, we might be tempted to simply multiply the number of months (t) by the new price ($35), but notice that will not take into account the fact that the first 12 months cost $10.
 We have to account for the change in cost by still charging $10 for the first 12 months, then change the price only for the remaining months.
 We have t−12 remaining months which will get the increased price. Thus, if we buy more than 12 months, the cost comes out to
10 ·12 + 35(t−12)  To create the piecewise function, notice that the first pricing scheme occurs only when 0 ≤ t ≤ 12, while the second pricing scheme only happens for t > 12.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Answer
Piecewise Functions
Lecture Slides are screencaptured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
 Intro
 Introduction
 Analogies to a Piecewise Function
 Notations for Piecewise Functions
 Example of a Piecewise (with Table)
 Example of a NonNumerical Piecewise
 Graphing Piecewise Functions
 Graphing Piecewise Functions, Example
 Continuous Functions
 Interesting Functions: the Step Function
 Example 1
 Example 2
 Example 3
 Example 4
 Intro 0:00
 Introduction 0:04
 Analogies to a Piecewise Function 1:16
 Different Potatoes
 Factory Production
 Notations for Piecewise Functions 3:39
 Notation Examples from Analogies
 Example of a Piecewise (with Table) 7:24
 Example of a NonNumerical Piecewise 11:35
 Graphing Piecewise Functions 14:15
 Graphing Piecewise Functions, Example 16:26
 Continuous Functions 16:57
 Statements of Continuity
 Example of Continuous and NonContinuous Graphs
 Interesting Functions: the Step Function 22:00
 Notation for the Step Function
 How the Step Function Works
 Graph of the Step Function
 Example 1 26:22
 Example 2 28:49
 Example 3 36:50
 Example 4 46:11
Precalculus with Limits Online Course
Transcription: Piecewise Functions
Hiwelcome back to Educator.com.0000
Today, we are going to talk about piecewise functions.0002
So far, all of the functions we have seen use a single rule over their entire domain.0005
No matter what goes into the function, as long as it is part of the domain, the same process happens to that input.0009
For example, consider x^{2} + 3; the same process happens, no matter what input goes in.0015
While f(1), f(27), f(473) all produce different outputs, the function is doing the exact same thing to each.0022
It squares the input; then it adds 2; f(1) is 1^{2} + 3; f(27) is 27^{2} + 3; f(473) is (473)^{2} + 3.0034
Different inputs will make different outputs, but ultimately the process we are going through is the same for each one of these.0046
What if the rule wasn't always the same, thoughwhat if the rule we used, the process we used, changed depending on the input?0053
Instead of always using the same process, the transformation could vary based on what goes into the function.0061
This is the idea behind piecewise functionsfunctions that do different things over different pieces of their domain.0066
Let's start by looking at two analogies, to get this idea really into our heads.0073
The first analogy: imagine you are cooking potatoes, and you are going to turn them either into mashed potatoes or French fries.0078
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.0085
You just bought a big pile of potatoes, so you have some big potatoes and some small potatoes.0092
So, some of them are going to make better fries, and some of them would probably make better mashed potatoes.0097
So, the big ones become fries, and the small ones will become mashed potatoesOK, that makes sense.0101
When you start cooking, you begin to work through your pile of potatoes.0106
You pick up one potato at a time, and you start by deciding if it is small or if it is big.0109
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 friesit seems pretty simple.0114
But we are getting sort of that idea of piecewise functions here, because in both cases, something will happen to the potato.0122
The potato will go through a process, but the transformation is different, depending on the type.0128
Depending on what kind of potato we haveif it is big potato or if it is a small potatodifferent processes are going to occur next.0134
We have to pay attention to what we are putting into our cooking process before we know what steps to take next.0141
Another onehere is another analogy: let's consider a factory.0148
Imagine this factory where, depending on what materials you bring to the factory, it produces different things.0151
If you bring along two kilograms of wood, they make a chair.0157
If you bring in 400 kilograms of metal, they make a car.0160
2 kilograms of woodthe factory makes a chair; if you bring in 400 kilograms of metal to the factory, it makes a car.0165
Now, notice that it is not enough to simply say how much material you bring.0172
If you bring in a mass of 400 kilograms, that is not enough information.0176
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.0181
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.0191
400 kilograms isn't enough information; we need to know 400 kilograms and what type it is.0198
We need to know what category it belongs to.0202
So, something is going to happen to the material, no matter what; but we have to know the category that the material belongs to.0205
We have to know what we are putting innot just a specific number, but what sort of group it is from, before we can tell what is going to happen.0210
This is now a vague sense of piecewise functions; we have this idea that a piecewise function0219
is something where a process will happen to the input, but different things will happen, depending on the specific nature of the input.0225
That is a really good sense for what a piecewise function is.0232
So now, let's consider the notation that is used for piecewise functions.0234
Generally, it is f(x) = [...and this bracket just says that it breaks into multiple different things.0237
There are different possible paths that we can take.0243
So, transformation rule 1 is our normal rule, like x^{2} + 7; and then it says x is in category 1.0246
So, what that means is that we look at our input; and then we go and we look at the various categories we have.0253
Is x wood? Is x metal? Is the potato big? Is the potato small?0262
We look for which category it belongs to; once we have found that it belongs to category 2,0268
then we go ahead and use transformation rule 2 on that input.0275
We look at the input that is going into the function; we then see which one of the categories it belongs to.0281
And that tells us which of the rules to use.0287
When we want to talk about the function, we have this bracket, so we can see all of the possible rules at once,0289
and all of the categories that go along with the possible rules.0294
Which set of circumstances do we use each rule under?0297
Given some input, we first check which category it belongs to.0301
Then, we use the corresponding transformation; so there are all of these transformation rules; we first check the category; then we use it.0306
Also, notice that since f is a function, two categories cannot overlap.0313
So, categories cannot overlapwhy? because, if they overlapped, and they had different rules,0320
we would get two different outputs from using the same input.0326
We would get two different outputs from using two different rules, if the categories overlapped.0330
Remember, if we put x into a function, it has to only produce one output.0335
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.0341
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,0347
or we would have to make sure that the categories don't overlap.0353
We are allowed to have categories overlap; but if that is the case, we have to make sure that the transformation rules0356
produce the same output during that overlapping space; otherwise, we have broken the nature of being a function.0361
All right, let's start looking at some examples to help us get a sense of how to use this notation.0367
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.0372
First, our potatoes analogy: potatoes of input...how does our potato function work on input?0378
If we plug in x, the first thing we do is see, "Is x small?"0385
If x is small, then we turn it into mashed potatoes; if x is big, then we turn it into French fries.0389
So, we plug in our potato x, and then we see which category x belongs to.0396
The same sort of thing is going on at the factory.0403
If we plug x, our input, into the factory function, some sort of quantitysome number of kilogramsof a material,0405
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.0414
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.0422
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.0435
Great; all right, let's see an example of the piecewise function, actually working through with numbers.0443
Here is a table; this is the most extreme sort of table we can use.0449
When we are actually doing this, we probably won't want to use a table that has this much possible information in it.0453
But we will get the idea of how piecewise functions work from this table.0459
So, to start with, let's look at what would happen to 4.0463
Well, actually, first let's look...which one would 4 belong to?0467
4 would be in x < 1; so it is going to belong in x^{2}  1, and it is not going to belong in the 2 rule,0471
because 4 is not between 1 and 1, and 4 is not greater than 1.0480
So, it is going to knock out these two rules.0488
Next, 3: 3 is still less than 1, so once again, that knocks out the second rule and the third rule.0491
What about 2? Well, once again, 2 is still less than 2, so that knocks out the second rule and the third rule again.0499
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.0506
1 is less than or equal; it knocks out that first rule, but 1 is still not greater than 1 from our third category.0514
So, it knocks out the third rule, as well.0523
What about plugging in 0? Well, 0 is not less than 1; and 0 is not greater than 1;0525
so our first and third categories just got knocked outthe first and third rules are out.0532
What if we plug in 1? Well, once again, 1 is equal to 1, so it is part of this second category.0535
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.0541
We get to 2, and finally 1 is less than 2; 2 is greater than 1; we have 1 being less than 2,0550
so we are using the third rule, which means that our first rule and our second rule are knocked out.0558
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.0565
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.0575
So now, we have a sense of how this table comes together.0585
So now, let's actually start plugging in numbers.0588
4 goes into x^{2}  1: (4)^{2}  1 gets us +16  1, so we get 15.0590
What about 3? We plug in (3)^{2}  1; that gets 9  1, so we get 8.0602
What about 2? We plug in (2)^{2}  1; 4  1 gets us 3.0611
All right, now we switch rules; for this one, we plug in 1, but 1 doesn't really do anything.0620
All the function says is that, if you are between 1 and 1 as your input, it outputs 2.0627
It doesn't care what you are putting in as an input, as long as it is between 1 and 1.0632
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.0636
1, 0, and 1; it is 2...it is going to be a constant value of 2 in that interval.0644
Now, we switch rules once again, and we are at 2, 2(2) + 4; 2(2) gets us 4; 4 + 4 gets us 0.0649
What about 3? 2...plug in our 3...+ 4; 2(3) gets us 6; 6 + 4 gets us 2.0660
Plug in 4: 2(4) + 4...2 times 4 gets us 8; 8 + 4 gets us 4.0670
We have managed to fill out this table.0679
The important thing is to start by figuring out which one of these inputs is going to go to which category.0681
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.0688
All right, let's also see an example of a nonnumerical piecewise.0694
Many lessons ago, when we first introduced the idea of a function, we talked about a nonnumerical initial function.0698
It took in names spelled with the Roman alphabet, and it output the first letter of the name.0704
For example, if we gave it the name Robert, the initial function would come along,0708
and it would say, "Your first letter is R, so we put out the letter R; done!"0715
So, it is just going and saying, "Let's grab the first initial and do that."0722
That was our idea of the initial function when we first introduced it.0725
We can have functions operating on nonnumerical things.0728
But we can also have piecewise functions on nonnumerical things.0730
We can modify that and make a piecewise function; we will have f(x) is equal to two categories.0734
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).0740
It is the first letter of the name, if x starts with A  M.0749
And then, it is the last letter of the name, if the name starts with N  Z.0752
And notice that that covers all of the possible letters that names could start with: A to M, N to Z;0758
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.0763
Albert: we plug in Albert, and Albert belongs to the red category; it belongs to starting with A to M.0771
That one is pretty easy; we use the first letter, so it gets A as the letter out of it.0778
What about Isabella? Well, Isabella is between A and M (A, B, C, D, E, F, G, H, I, J, K, L, M);0784
so Isabella also belongs to the red category; so it is going to return an I.0794
What about Nicole? Nicole is an N, so it is using the blue category; so it uses the last letter of the name.0800
The last letter of Nicole is E.0808
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).0812
So, we are going to be using the blue category, the second category where you use that last letter, T.0822
What about Zach? If we have Zach, Zach is pretty clearly going to be starting with a Z.0827
That means we are in the second category; we are going to cut off that last letter.0832
We are going to get the last letter, and it will be H.0836
We can have a piecewise function operating on nonnumerical things.0838
The idea of a piecewise function is just that we take in things; we see which category the thing belongs to;0841
and then we apply a rule based on the category it belongs to.0848
We figure out the category; then we apply a rule based on it; great.0851
Graphing piecewise functions: how do we graph these things?0857
It is actually really similar to graphing a normal function.0859
A series of points, (x,f(x)): when we graph x^{2}, the reason why (0,0) is a point0862
is because, if we plug in 0, we get 0; (0,f(0)); for x^{2}, we get a (0,0).0868
Then we plug in (1,f(1)), because we plug in 1, and it gets 1^{2}; so it is (1,1).0876
We plug in 2, and we get f(2); so that would be 2; and then 2^{2} is 4, so we would be at (2,4).0883
We are moving to a height of 4; and that is how it is working for x^{2}.0890
And that is how it works for graphing any function; we put in the input, and then we see what output that gets to.0893
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.0898
So, often it will look to use like the graph is changing at switchoversthat we are breaking from one thing to another.0905
And in a way, we are: we are switching between rules.0911
So, when x switches from one category to another, the shape or location of the curve can change,0914
because all of a sudden, we are doing a new way of outputting things.0919
It is not that nice, smooth connection anymore, because we are working all through x^{2}.0922
All of a sudden, we are jumping from x^{2} to maybe x^{3}  1.0926
And we are going to see, suddenly, a new, totally different kind of output when we change categories.0929
An important graphical note is that, to show inclusion, we use a solid circle.0934
Solid circles say "this point is herethis point is actually being included."0940
We show exclusion, excluding it, saying it is not there, with an empty circle.0946
Empty circles give us exclusion; exclusion is an empty circle; inclusion is a filledin circle.0952
That way, when we have two categories, like x < 1 and 1 ≤ x, we want to be able to know which curve owns x = 1.0961
And because it is less than or equal to, we would have 1 ≤ x, so it would get the dot at that point.0970
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.0977
Let's see an example: if we graph f(x) = x^{2}  1 when x < 1, 2 when 1 ≤ x ≤ 1,0985
and 2x + 4, which is when 1 < x, we would see this graph right here.0996
And I also just want to point out that that is what we just did in our tableour really big table,1001
where we figured out all of the possibilitiesthat is what we just did a few slides back.1004
So, x^{2}  1; we see this portion (I will make it in blue, actually) of the graph right here,1009
because x^{2}  1 is a parabola; we are seeing the left portion of the parabola, because it is only1018
the portion of the parabola when x is less than 1.1023
We plug in those values, and we see where it gets mapped.1026
However, because x is less than 1, we have exclusion right here; we are not allowed to actually use that point.1028
It gets right up to it, but it can't actually touch x = 1, because it has exclusion on x < 1.1036
And it is strictly less than; it is not actually allowed to equal 1.1044
Similarly, we have to flip to inclusion on the rule of being 2; so it gets to include it on this.1048
And then, it is just a constant from 1 up until 1; so we have this nice straight line here.1055
And then, in green, 2x + 4: 1 < x; we have this thing right here.1062
Now, why don't we see a dot at this junction right here?1069
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.1076
2x + 4, evaluated at 2...sorry, not 2, but 1, because we are at the point 1; 1 is our changeover, right here;1084
1 < x and x ≤ 1; so 2...plug in 1; plus 4; we get 2 + 4; so we get 2.1095
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.1107
It is sort of a hollow for 2x + 4; but then, it immediately gets filled in for 2.1116
So, ultimately, we don't see a break there.1123
There is no loss, because while one end is excluding it, the other one is including it.1125
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.1129
And that is one example of graphing a piecewise function.1135
All right, this is a great time to bring up the idea of a continuous function.1138
We are going to occasionally refer to the idea in this course, and it is going to come up a whole bunch in calculus.1142
So, it is important to get used to it now.1146
It is hard to formally define continuous right now; right now, we don't have enough symbolic technology.1148
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.1154
But we can understand what it means graphically; it makes great intuitive sense in pictures.1160
All of these different thingsall of these three different ways of talking about itall mean the exact same thing.1165
So, if a function is continuous, all of the parts of its graph are connected.1170
Its graph could be drawn without ever having to lift your pencil off the paper.1176
And there are no breaks in the graph.1181
These three things all mean the exact same thing.1183
The parts of the graph are connected; that means that there are no breaks.1185
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.1188
They are three different ways of thinking about it, but they all mean the same thing: the whole thing is connected.1194
It is a continuous flowthis nice, connected piece of information.1198
Great; let's look at some examples to help us understand this.1203
For a function to be continuous, it must be any one of the below statements (we just said them; let's say them again).1206
The parts of its graph are connected; the graph could be drawn1211
without ever lifting your pencil or pen off the paper; and there are no breaks in the graph.1213
So, here is one example of something being continuous.1218
Even though it has this sort of "juke" (it suddenly changes direction there) in this corner,1221
it is still continuous, because the graph connects in this corner.1228
The two ends touch; you can draw it in one smooth thing without ever having to take your hand off of drawing.1232
You could draw it in one smooth thing, without ever having to lift up.1239
So, it is a continuous graph, at least in that viewing windowwhat we can see.1243
Now, here is an example of something being not continuous.1247
This one right here is not continuous, because we have this break; all of a sudden, we jump locations.1251
We have an empty circle here, and we are now, all of a sudden, in a totally different place.1257
If we were to try to draw this, we could get up to here; but to go any farther,1261
we would have to lift our pencil up, move down, and now start down here.1265
So, we would be in a totally different place; there is a break in the graph.1269
The parts of the graph are not connected; it is not just one nice, connected curve; it is not a continuous function.1272
Finally, another one that not continuous: this one is pretty close to being continuous.1278
We can draw and draw and draw and draw and draw, but there is an empty point here.1284
We have one point; we have this single discontinuity, this single point that has been moved off of the line.1288
We have to move down here to draw in this single point; and then we go back to normal.1294
So, it is really, really, really close to being continuous; but it is not perfectly continuous,1298
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.1302
So, a continuous functionall of the parts of its graph are connected.1309
And that means even one point could be out, and it would break the continuity.1312
It would break being continuous; it would no longer be connected.1316
Great; that is an interesting function.1320
At this point, it is probably becoming clear that you can have some kind of weirdlooking functions.1322
So far, over the course of algebra and geometry, you have seen pretty reasonablelooking things, like x^{2}, √x, x^{3}...1327
Even the weirdest things you have seen have been pretty reasonablejust sort of smooth curves.1334
But we are starting to see, with piecewise functions, that things can be a little odd.1338
Let's look at another one, a little rulea rule that is a little more complex than x^{2},1344
something that is a little more interesting than the ones we have been used to so far.1349
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).1353
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.1360
The greatest integer less than or equal to xwhat does that mean?1370
The greatest integer less than or equal to xlet's try it out.1375
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.1377
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,1387
since there is 1, 0, 1, 2, 3; 3 is the greatest one that is less than or equal to 3.1393
What if we went higher, if we said 4? Well, 4 would be greater than 3; so it is not in the runningit doesn't have a possibility.1402
So, that would be 3; but what if we tried something that wasn't just already a straight integer, like, say, 4.7?1410
If we plugged in 4.7, well, what is the greatest integer that is less than or equal to x?1417
3 is a possibility; 4 is a possibility; 5 is a possibility; and it would keep going in either direction.1423
Well, if we went from the left, let's start like this; we could say, "11 is a possible thing! Let's go with 1!"1429
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.1444
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.1450
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.1456
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.1465
So, that means 5 is out of the running, and also anything larger than 5.1478
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.1481
And everything 5 or greater isn't going to work; so that means our answer is 4.7.1487
It is basically always rounding down; 4.7 would become 4; 3.5 would become 3; 2.5 would become 3,1492
because we have to round down, and what is below 2.5? 3.1501
All right, and finally, we can also sometimes call this the int(x), the integer function on x.1507
Sometimes you will see it denoted as that; it will be written as int(x), as opposed to [[x]].1514
It is the same idea thoughthis greatest integer thing, this step function where we are breaking.1520
Now, why is it called a step function? We will look at a picture, and that will help explain it a lot.1525
So, the graph of f(x) = [[x]] looks like this.1529
Why does it look like this? Well, remember, at 3, where do we get placed?1533
Well, 3 is an integer, so it just goes right here.1536
Well, what about anything in the middle? Anything in the middle would get placed onto 3,1540
because they would have to be rounded down to the greatest integer they are connected to.1545
So, that is what we get there.1548
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.1549
And so, it is going to have the same sort of thing; anything in the middle would end up getting placed onto 2.1555
But once we get to 1, we make it to this one; and so on and so forth.1560
And so, we just keep stepping along and stepping along and stepping along.1563
And every time we hit an integer, we jump up to the next height, and so on and so forth.1567
And so, we have the greatest integer less than or equal to x, which ends up looking like a staircase,1572
in terms of its steps, because we keep stepping up every time we hit a new integer.1577
Cool; all right, we are ready for some examples.1582
So, the first one just to get started: let's evaluate this function at four different points:1584
f(x) = 3x + 10 when x < 2, 8 when x = 1, and x^{2}  10 when x > 1.1589
All right, at f(3), first what we have to do is say, "Which category do you belong to?"1597
Well, 3 is less than 1, so it belongs to the 3x + 10 rule.1603
So, we use 3x + 10; we plug in the 3; we have 3(3) + 10; 9 + 10; so we have 1; f(3) = 1.1610
Great; what about f(1)what does that belong to?1624
Well, 1 = 1, so it is using this category right here, so that means we have 8; so we have 8.1631
There is nothing else that we have to do; it is already as simple as it can be; f(1) = 8.1638
And that is our answer, right there.1642
What about f(0.9)? This one is really close to 1; but remember, this thing was x = 1, and only happens on precisely 1.1645
0.9 is, in fact, slightly greater than 1; 0.9 is greater than 1, so we use the rule x^{2}  10.1655
We plug in that 0.9; we have (0.9)^{2}  10; 0.9 squared is 0.81; it becomes positive.1666
Anything squared becomes positive, as long as it is a real number.1676
0.81  10 becomes negative, because the 10 is bigger: 9.19.1679
So, f(0.9) is equal to 9.9.1688
Finally, one more example, f(5); which category does this belong to?1694
It pretty clearly belongs to x > 1; 5 > 1, so we use the x^{2}  10 rule once again; we use that process.1699
Plug in the 5; 5^{2}  10 is 25  10, is 15; so f(5) = 15.1707
And there we are; and that is how you evaluate a piecewise function.1720
You see which category it belongs to; then you plug it into the appropriate rule, and you just plug it in and work,1722
like you are doing a normal function at that point; great.1727
The next one: all right, in this one, we will graph a piecewise function; so our function this time is:1729
f(x) = x + 6 when x ≤ 3 and x^{2}  2x + 1 when x > 3.1734
So, first, let's make a table to help us graph this thing.1742
x and f(x); what would be a good place to start out?1746
Well, we have 3 showing up here and here; so that is probably going to be the midpoint, mid"zone" in our graph.1755
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.1762
Let's try plugging in...which rule will we end up using?1778
Well, when x is less than or equal to 3, we will end up using the things that are above 3 or equal to 3.1781
So, this rule up here gets the x + 6 portion; and down here, when we are below the line, we get x^{2}  2x + 1,1791
because then x is greater than 3; 2 is greater than 3; 0 is greater than 3; etc.1805
All right, so let's try doing some of these.1811
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?1813
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 ≤ 3.1830
Now, what about going the other way?1844
Well, if we plug in 2 into x^{2}  2x + 1, we have (2)^{2}  2(2) + 1.1845
Let's plug in all of them, and then we will just do them at once.1855
1^{2}  2(1) + 1; 0^{2}  2(0) + 1; 1^{2}  2(1) + 1; what do these all come out to be?1857
Well, first, 2 squared becomes positive 4; so we hit that with another negative, and we have 4 right here.1874
2 times 2 gets us +4, + 1, so 4 + 4 gets canceled; and then + 1...we get 1.1882
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 2.1892
0 squared gets us 0; 2 times 0 gets us 0, plus 1we get 1.1903
1 squared gets us 1; 2 times 1 gets us 2, plus 1; so we get 2 here.1909
Great; all right, at this point, we can start graphing this thing.1920
We are graphing from 6 to 1; and our extreme yvalues are...we have from 0, 1, 2, 3, so we will make it 1, 2, 3, 4, 1, 2, 3, 4, 5.1924
That is probably enough information; I have to do down there...1, 2, 3, 4, 1, 2, 3, 4, 5.1947
And there doesn't seem to be any reason why we shouldn't do this on a square axis.1957
So, the tick mark length, the length of our vertical tick marks, can be the same as our horizontal tick marks.1961
And of course, I am just doing this by hand, so it is approximate.1966
But this isn't too bad: 1, 2, 3, 4, 5, 6, and it would keep going out that way, as well.1968
All right, 1, 2, 3, 4, 5, and 6; positive 1, positive 2; great.1977
So, at this point, we plot down our points, just like we are doing a normal thing.1985
6 goes to 0, right here; 5 goes to positive 1 here; 4 goes to positive 2 here; 3 goes to positive 3 here.1988
And at this point, we have the line portion.2003
Does the line keep going to the right, though? No, because it stops once it goes greater than 3.2005
It only works, the rule only happens, when x is less than or equal to 3.2013
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."2016
Now, what about the parabola part of it?2024
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.2026
So, we have a pretty clear parabolic arc going on here.2039
We are used to this; and it is going to keep going straight off forever to the right,2045
because it is x > 3; so as long as we are continuing to go to the right, it will continue on.2049
What happens to the left, though? We know what is happeningit is going to be in a parabolic arc.2054
But we are not quite sure where it is going to land, because it has to stop somewhere.2058
But we don't know what height it will stop at.2062
We know it will stop just before 3; so it will stop at 2.9999999999999...forever, continuing forever and ever.2065
It can't actually touch 3, but it can get infinitely close; it can get right up next to it.2074
So, let's figure out where it would be going if it got right up next to it.2078
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.2082
Now...well, not quite forever, because then it would turn into 3.2093
But the point is 2.lotsofnines; now, I don't know about you, but I don't want to have to plug in 2.9999999 into a calculator,2096
because it is going to end up getting me these ugly numbers, and I will end up having decimals.2104
And really, when you get right down to it, isn't 2.99999999 going to behave a lot like we plugged in 3?2106
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.2113
We will just know that it will be an empty circle of that, because it has exclusion; it has strictly greater than.2120
So, we will plug in 2.99999999 and 9, which still belongs to the x^{2}  2x + 1,2126
because 2.999999999 is greater than 3, if only by a little tiny bit.2132
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 3.2137
So, (3)^{2}  2(3) + 1what does that come out to be?2144
(3)^{2} becomes 9; 2(3) becomes +6; plus 1, so we have 9 + 7 = 2.2151
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 3.2163
It is going to be hollow there, because we are excluding itit is not actually allowed to go to that point,2170
because the exclusion was already put on that first category on that first rule.2175
This will curve down in a parabolic arc into the exclusion hole, and then just stop right there.2180
It doesn't actually get to touch 3, but we can basically calculate it as if it had gotten to 3,2185
because 2.999999999999 is so close to being just like 3, we can calculate it as if it had gotten there.2190
But then, we just have to remember that we have to make sure that we put it in this circle here,2198
because we are actually excluding it at 3, because x isn't equal to 3 if x > 3.2202
Great; all those ideas that we just talked about are going to come up a lot with this.2209
Let's go just a little bit off and pretend that we are using the real number, and then see what it is like.2214
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.2219
And we will get an idea of "Oh, that is why we can do this sort of thing."2226
So, once again, we will set this up in the same way: (x,f(x)): now what values...2229
Clearly, 2 is kind of important; it shows up in a lot of places.2239
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.2244
First, let's do the domain; how do we come up with the domain?2251
Well, remember, domain is all of the inputs that are allowed to go into a function.2256
x^{2}  5 never breaks down; 3 never breaks down; 2x + 1 never breaks down; so none of the rules break down.2261
So, none of the processes, none of the rules, break; they are always defined.2272
However, are the categories always defined?2281
x < 2 means we can just keep on going; we can keep on going.2286
So, it is really negative infinity less than x; so we can go all the way down to negative infinity.2290
What about to the right, though? Is there anything that we are not allowed to get to?2295
Well, negative infinity up to 2; and then 2 is here at equal; and then 2 is less than...2298
so we have covered all of our bases, from negative infinity up to 2; and keep going, up until 1.2302
Are there any rules for what happens if x is greater than 1?2309
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 > 1.2313
So, no rules for x > 1 means that f doesn't tell us what to do if we are plugging something in.2324
The f fails to tell us what to do to this input if we plug in something that is greater than 1.2334
If we plug in, say, 500, we look at this, and we say, "Oh, this doesn't belong to any categories."2340
So, f is undefined at 500; it doesn't work; it is not in the domain.2345
So, the domain fails to contain everything in x > 1; so that means our domain is not going to be x > 1.2350
That is the things of failure, because we don't have rules; the domain is everything from negative infinity2362
(we use a parenthesis for negative infinity, and infinity), and we go up until 1; and we include the 1,2367
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.2373
So, f has a domain from negative infinity up until 1, including 1.2378
Great; now, let's build up that table.2383
2 seems like a good place to make our middle; and if we are above 2, we will use which rule?2385
We use the x^{2}  5 rule.2394
Sorry, by "above," I meant to say more negative than 2.2399
And if we are below on this table, which is to say more positive, closer to 0, we are going to use 2x + 1.2402
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 know2411
where it is going if it had been allowed to get to 2.2423
So, for the above part, we will say 2.0001; and 1.9999; these things are because .1999999 is greater than 2,2426
and 2.000001 is less than 2, but they are going to behave effectively as if we had plugged in 2.2440
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.2448
All right, the first one, 2: what are our f(x)'s?2454
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 2.2459
So, it automatically gives out 3 at 2.2466
What about 2.00001, which would use the x^{2}  5 rule? Well, that is about the same thing as plugging in 2.2471
We have (2)^{2}  5; keep goinglet's just keep going up to get them all written out.2477
(3)^{2}  5, and (4)^{2}  5what do those all equal?2484
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 5.2491
And (4)^{2} is +16; 16  5 is 11; OK.2502
What about the other way, if we go to the 2x + 1 rule?2506
Well, if we had plugged in 2, we would get 2 times 2; we aren't literally plugging it in.2510
We are just saying, "What if we had gone all the way up to it? Let's see what would have happened,"2514
even though ultimately we will have to exclude it, because we have these strictly less than and strictly greater than signs.2519
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...2531
2 times positive 1 (I got that confused with the one above it); 2 times 1 plus 1; what do those all equal?2543
The thing that is effectively going to be like 2...2 times 2 is positive 4; 4 plus 1 is 5.2551
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 1.2557
Great; so now we are in a position to be able to graph it.2567
Our extremes...vertically we can get up to really high things when we are in the x^{2}  5; so we won't worry about the 11 part.2569
But we are going between extremes of 5, maybe a little lower; so we will graph this...2575
We never get to very low values, it seems; so we will put our corner down here.2585
And we also never get past 1; remember, our domain is only ∞ up until 1.2589
So, we also don't have to have a whole lot of stuff on the right.2593
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, 5.2597
Great, and that is plenty of room, because we only get up to 4; and we know that x^{2}  5 is going to blow out.2622
So, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5; making tables...making axes.2627
So, let's plug in some things and see what happens.2643
We plug in 4, and it goes out to 11; so we can't even see it; it is so high up.2646
3 gets to 4, though; we can definitely plot that; so 3 goes to 4, (3,4).2651
2, if it had a 2...it doesn't actually have it, but we know that 2.0001 would practically be going to 1.2659
So, we are putting an exclusion hole down here, just below the 2.2670
At 2, though, we actually end up being at 3; so we have this single point right there.2676
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.2683
1 is at 3; 0 is at 1; 1 is at 1; and it stops right there, because we stop at 1.2693
We can't go any higher than positive 1; our domain caps out there.2704
So, our straight line is just a straight line, up until where it stops at that exclusionary hole.2709
We have this point in the middle, the 3 point, but it is just x = 2; and then we have a parabola curving up.2715
It is already moving pretty fast by this point, so it is not going to be a nice, smooth parabola like this.2722
It is already moving fast up, because it is pretty far up.2727
It manages to jump from 1 to 4, and then from 4 to 11; so it is not the bottom part of the parabola.2731
It is already in process, in a way; so curve this parabola up, and it just zooms way, way off really quickly.2735
All right, that is basically what we are seeing here for our graph.2745
We have a parabola on the lefthand side, which drops to the single pointthere is just a single point in the middle.2747
And then, we switch to 2x + 1, which goes to 1, and then stops at 1,2754
because the category just stops at 1; so it stops right here, and we don't have anything farther to the right.2760
There is nothing further off to the right, because the categories don't include anything further right.2765
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.2771
After 200 minutes, they charge 5 cents for each additional minute.2780
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.2784
So, t is the minutes spoken.2791
It is pretty easy for us to figure out what the first part is.2793
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.2796
p(t) =...well, a $20 flat ratethey charge us $20, and then they charge us 10 cents for each additional minute.2812
So, $20 plus that additional 10 cents...how many minutes did we have? We had t.2820
So, 0.1tthat is what it is; let's do a really quick testlet's say if we had talked 100 minutes.2825
Then 100 minutes times 10 cents would be $10, so we would have a $30 total,2834
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 30.2839
Great, so the first part of it checks out.2846
What about the second portion, though?that is where things start to get a little complicated.2848
So, in the second portion, when we are over 200, which is to say t > 200...2853
and actually can be greater than or equal, because we know that they are going to have to agree;2861
there is not going to be a sudden jump there; and we will talk about that more later.2865
It is a way of checking this function, actually.2867
We know that it is going to be 5 cents for each additional minute.2870
Our first thought might be, "Oh, great, easy; it is going to be 0.05t."2875
Not truethis is not going to be the case.2880
Why not? Because it is for each additional minute, over 200; so after 200, you get charged at 5 cents per minute.2884
Before that, you still get charged at the 10 cents; so how many minutes over 200?2894
Well, that is not too hard; we know that we have t minutes total.2904
We know that we are already over 200, so it is going to be the number of minutes we have talked, minus 200.2908
So, t  200 is the number of minutes we talked; so it is 0.05 cents, times t  200.2914
Now, that is the amount of additional money that will be on top of some lump.2922
How much is it to even make it to 200 minutes in the first place?2927
Well, 200 minutes in the first place: let's see what it is from our first one.2930
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 40.2934
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.2947
So, for the second portion, our function is going to be p(t) = $40, the lump sum that we have to pay at first2967
to have even made it to the 200minute mark, plus 5 cents for the number of minutes over 200 minutes.2975
So, our function has been broken into 2 pieces; so we have a piecewise function here, p(t) = 20 + 0.1t when t ≤ 200,2984
and 40 + 0.05 times the minutes over 200 when t ≥ 200.3001
Now, we know that the two have to agree; otherwise, people would make sure to make that jump or not make that jump,3014
because otherwise with the sudden change, or the switchit wouldn't make sense for the phone company3019
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.3024
It is going to just continue in a continuous function, we would expect.3029
So, we can check this; and we can check and make sure that, indeed, p(200) = 20 + 0.1(200).3034
We already did this before; it was $40; and let's check and make sure that the second portion, p(200), would agree.3044
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 40.3053
So, those two things check, and our function, price in respect to time, makes perfect sense.3068
So, p(t) = 20 + 10 cents per minutes when minutes are less than or equal to 200,3075
or 40 + 5 cents per minutes over 200, when the number of minutes is greater than or equal to 200.3080
Great; I hope piecewise functions are making a lot more sense now.3087
Remember: it is an idea about putting into the category, then applying the rule based on the category.3090
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.3095
All right, we will see you at Educator.com latergoodbye!3100
1 answer
Last reply by: Professor SelhorstJones
Sun Aug 28, 2016 2:22 PM
Post by Hitendrakumar Patel on August 24, 2016
Why are these new piecewise rules implemented?
1 answer
Last reply by: Mohammed Jaweed
Sun Jul 12, 2015 4:56 PM
Post by Mohammed Jaweed on July 10, 2015
In example 2, for the bottom potion, how did you end up with 1 from 1^2
1 answer
Last reply by: Professor SelhorstJones
Fri Aug 29, 2014 6:35 PM
Post by John K on August 23, 2014
Professor,
In the graphing piece wise function example(16:26), for f(x)=2, why did we use inclusion at the point 1?
1 answer
Last reply by: Professor SelhorstJones
Fri Oct 4, 2013 6:08 PM
Post by Min Journey on October 4, 2013
@ 7:24
I thought the condition for 2x+4 is that 1<x. But 4 is not greater than 1. Or is this only after you plug in the xvalue?