For more information, please see full course syllabus of Pre Calculus

For more information, please see full course syllabus of Pre Calculus

## Discussion

## Study Guides

## Practice Questions

## Download Lecture Slides

## Table of Contents

## Transcription

## Related Books

### Function Petting Zoo

- An important step in learning mathematics is becoming accustomed to how various functions behave and what their graphs looks like. While long-term experience is the surest way to familiarize yourself with functions, this lesson will help give you a head-start on developing your "function intuition".
- Other teachers or books might call a similar lesson a
*Function Library*or*Parent Functions*. The point is the same, though: to introduce/review a wide variety of fundamental functions, along with their characteristics and graphs. - As we work through these functions and see their graphs, don't forget: axes matter. Pay attention to the "graphing window" so you can interpret what you're seeing. For the majority of these graphs, we'll use x: [−10, 10], y: [−10, 10], but a few graphs will be different.
- We can't really explain or show the graphs here in the notes, so make sure to check out the video if you need to familiarize yourself with these fundamental functions. Here is a list of the functions we examine in this lesson:
- The
*Constant*Function: f(x) = k - The
*Identity*Function: f(x) = x - The
*Square*Function: f(x) = x^{2} - The
*Cube*Function: f(x) = x^{3} - The
*Square Root*Function: f(x) = √x - The
*Reciprocal*Function: f(x) = 1/x - The
*Absolute Value*Function: f(x) = |x| - The
*Trigonometric*Functions: f(x) = sin(x) & g(x) = cos(x) - The
*Exponential*and*Logarithmic*Functions: f(x) = 10^{x}& g(x) = log_{10}(x)

- The

### Function Petting Zoo

- Compare it to the library of parent functions you learned in the lesson.
- This has a smoothly curved shape where both ends point in the same direction. As it gets further to the left or right, it becomes steeper. These qualities taken together tell us which function it is.

*square*function: f(x) = x

^{2}

- Compare it to the library of parent functions you learned in the lesson.
- This has two distinct lines coming together into a point. As it gets further to the left or right, the steepness of the line does not change. These qualities taken together tell us which function it is.

*absolute value*function: f(x) = |x|

- Compare it to the library of parent functions you learned in the lesson.
- This has two curves coming out in opposite directions. As it gets further to the left or right, the steepness of the line increases quickly. These qualities taken together tell us which function it is.

*cube*function: f(x) = x

^{3}

- Compare it to the library of parent functions you learned in the lesson.
- This is a flat line. As it gets further to the left or right, it continues to stay totally flat. These qualities taken together tell us which function it is.

*constant*function: f(x) = k

- Compare it to the library of parent functions you learned in the lesson.
- This has a single curve coming out in one direction. As it gets further to the right, the steepness of the curve decreases quickly. These qualities taken together tell us which function it is.

*square root*function: f(x) = √x

- Compare it to the library of parent functions you learned in the lesson.
- This is a line. It is not constant, though, because it slopes. These qualities taken together tell us which function it is.
- We would normally call this a
*linear*function, but in this case, it comes from the parent function of the*identity*function: the most basic of linear functions.

*identity*function: f(x) = x

- Compare it to the library of parent functions you learned in the lesson.
- This has a smoothly curved shape where both ends point in the same direction. As it gets further to the left or right, it becomes steeper. These qualities taken together tell us which function it is.

*square*function: f(x) = x

^{2}

- Compare it to the library of parent functions you learned in the lesson.
- This has a curved shape which flies off to ±∞ in one location. As it gets further to the left or right, it becomes flatter. These qualities taken together tell us which function it is.

*reciprocal*function: f(x) = [1/x]

- Compare it to the library of parent functions you learned in the lesson.
- This has two curves coming out in opposite directions. As it gets further to the left or right, the steepness of the line increases quickly. These qualities taken together tell us which function it is.

*cube*function: f(x) = x

^{3}

- Compare it to the library of parent functions you learned in the lesson.
- This has two distinct lines coming together into a point. As it gets further to the left or right, the steepness of the line does not change. These qualities taken together tell us which function it is.

*absolute value*function: f(x) = |x|

*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

### Function Petting Zoo

Lecture Slides are screen-captured 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
- Don't Forget that Axes Matter!
- The Constant Function
- The Identity Function
- The Square Function
- The Cube Function
- The Square Root Function
- The Reciprocal Function
- The Absolute Value Function
- The Trigonometric Functions
- The Exponential and Logarithmic Functions
- Transformations and Compositions
- Example 1
- Example 2
- Example 3
- Example 4

- Intro 0:00
- Introduction 0:04
- Don't Forget that Axes Matter! 1:44
- The Constant Function 2:40
- The Identity Function 3:44
- The Square Function 4:40
- The Cube Function 5:44
- The Square Root Function 6:51
- The Reciprocal Function 8:11
- The Absolute Value Function 10:19
- The Trigonometric Functions 11:56
- f(x)=sin(x)
- f(x)=cos(x)
- Alternate Axes
- The Exponential and Logarithmic Functions 13:35
- Exponential Functions
- Logarithmic Functions
- Alternating Axes
- Transformations and Compositions 16:08
- Example 1 17:52
- Example 2 18:33
- Example 3 20:24
- Example 4 26:07

### Precalculus with Limits Online Course

### Transcription: Function Petting Zoo

*Hi--welcome back to Educator.com.*0000

*Today, we are going to have our function petting zoo.*0002

*An important step in learning mathematics is becoming accustomed to how various functions behave and what their graphs look like.*0005

*While long-term experience is the surest way to familiarize yourself with functions,*0011

*and you have probably already gotten some of that experience (when you see x ^{2} show up,*0014

*you get a sense that you are going to see a parabola), this lesson is here to give you a head start on developing your function intuition.*0017

*Maybe we will see some that you haven't really talked about before; I am actually sure that you will see a couple that you haven't really talked about before.*0023

*And you will get the chance to develop it; we will talk about various properties, and that sort of thing.*0029

*We can just get a sense of "when I see functions in this type, I know to expect a graph like this; I know to expect certain kinds of behavior."*0033

*So, we are going to graph various fundamental functions; and we will talk briefly about their key points.*0041

*Don't worry about memorizing this information; it is not here because you are going to have to know it because you are going to be drilled on this.*0045

*It is never going to be tested directly, probably (I suppose there might be a couple of teachers out there who would test directly on it).*0053

*But really, what this is about is exposing you to these so that you are ready to understand things better down the road.*0059

*And other teachers or books might call a similar lesson a function library or parent functions.*0065

*I personally think it is kind of fun to call this a function petting zoo,*0071

*because really, we are just going out and meeting a bunch of different functions.*0074

*We are getting the chance to interact and play with one function, another function, another function...just for a little bit at a time.*0077

*The point of any of these names, though, is the same thing.*0083

*Function library, parent functions, function petting zoo--all of these things are just to introduce or review*0085

*a wide variety of fundamental functions, and then also to talk about their characteristics and their graphs,*0090

*to get you the chance to develop that function intuition, so that when you see a function,*0096

*you know what to expect out of it, even if you haven't worked with that precise one before.*0099

*All right, before we get into this, though, don't forget that axes matter.*0103

*Remember from our very first lesson on graphs: how the axes are set up has a huge impact on what the graph looks like.*0108

*Whenever you look at a graph, pay attention to how the axes are set up.*0115

*For consistency, and to help us see how various functions behave differently, all the graphs we are about to see will be on the same axes.*0119

*So, the graphs we are about to see are going to be on the same axes; we will see them on the x, horizontal, -10 to 10, and the vertical -10 to 10.*0125

*Notice that these are square axes; the length of the horizontal is the same as the vertical length.*0134

*So, it is going to not give any sort of weird curving to it; we won't be squishing the picture from its "natural size" or its "natural shape."*0139

*These axes will help us see how each one of these compares to the other one.*0147

*We will have a general template to understand how the shape of this one is different from the shape of this one, is different from the shape of this one.*0151

*All right, let's get going: the very first one is the constant function, f(x) = k, where k is a constant--just some constant number.*0157

*So, things to notice about this: for example, in this one, we have that k is a little bit more than 3.*0170

*The input doesn't matter; no matter what we put in, 2 gets mapped to the same thing as 10, gets mapped to the same thing as -8.*0175

*Whatever we put in, it all gets mapped to k.*0184

*Output is thus always the same; whatever we are putting in, once again, it doesn't matter what we put in; it always gives the same thing out.*0187

*And then finally, what does it put out? It always outputs a horizontal line at k.*0200

*We plug in 5; it gives out k; we plug in 20; it gives out k; we plug in -47; it gives out k.*0207

*When we look at all of that together, graphically we are seeing a horizontal line at height k; great.*0215

*The identity function, f(x) = x: it is called the identity function because whatever we plug in is what we get out of it.*0224

*The input is the same as output; if we plug in 7, we get out 7; if we plug in -47, we get out -47.*0232

*So, what we get out of this is this nice, straight line that just cuts perfectly between the x-axis and the y-axis.*0248

*We see a slope of m = 1 because of this: slope equals 1, because it has to cut evenly between them.*0257

*Otherwise, it wouldn't be giving an identity, where the same thing that goes in...if 6 goes in, then 6 has to come out.*0268

*Whatever goes in is what comes out--all right, that is the identity function.*0275

*The square function, f(x) = x^{2}: the first thing to notice is that, as it goes to the right...*0281

*notice how it starts to go up faster and faster; and as it goes to the left, it goes up faster and faster.*0288

*The height increase, how fast it is going up, its rate of change--height increase speeds up farther out.*0294

*The farther we get away from the middle, the faster it is moving up.*0304

*Height increase is speeding up farther out.*0309

*The other important thing to point out is that, on the right, we are going up; and on the left, we are going up.*0311

*On the square function, the ends go in the same direction; the ends of graph go in the same direction--they point in the same direction.*0318

*This is an important idea about the square function: we can trust the fact that it will cup in the same direction.*0335

*The right side goes off in the same way as the left side.*0341

*The cube function is similar to the square function: its height increase speeds up, the farther we get out.*0345

*In fact, its height increase is going to speed up even faster; when we get to 2 on x ^{2}, we are only on an output of 4.*0360

*But when we get to 2 on x ^{3}, we are at an output of 8, 2 times 2 times 2.*0367

*So, its height increase speeds up, and it speeds up even faster than it does on x ^{2}.*0372

*The other thing to notice is that, if we were to continue the graph, one side goes up, and the other side goes down.*0377

*The ends of the graph point in opposite directions; unlike in x ^{2}, where they point in the same direction,*0383

*the ends of the graph in x ^{3} point in opposite directions.*0390

*The ends of the graph when we are doing the cube function will point one way up and one way down--opposite directions.*0398

*All right, the next one is the square root function: for the square root function, notice:*0410

*the farther we go out, the slower it is increasing in height.*0416

*To get to a 2 of height, it has to put in a 4; to get to a 3 in height, it has to put in a 9; to get to a 4 in height, it has to put in a 16.*0420

*So, height increase slows down the farther we get out--height increase slows down.*0430

*Also, notice the fact that there is nothing over here; there is nothing on the left side of the graph,*0440

*because if we try to plug in a negative number, it is not in the domain; so negatives are not in the domain.*0445

*So, in a way, √x only looks like half of a graph, because it doesn't really keep going.*0460

*Part of it just stops, because if we try to plug in negative numbers, square root fails to work.*0465

*There is no number that you can square (no real number, at least) that will make a negative number.*0471

*A negative number, squared, gives you a positive number; a positive number, squared, gives you a positive number.*0477

*So, there is no number that you can square that will give you a negative number, at least in the real numbers.*0481

*So, the negatives are not in the domain of the square root function.*0487

*The reciprocal function, 1/x, is called the reciprocal because the reciprocal of a number is just one over that number.*0492

*So, in this one, height increase, as we get farther and farther out, slows down.*0498

*Height increase slows down when we are far away from the y-axis.*0506

*The farther we get from the y-axis, the slower the height increase becomes.*0518

*And this makes sense, because at 2, we are at 1/2; at 4, we are at 1/4; at 6, we are at 1/6; so the height increase...*0525

*well, I guess decrease--it increases as we go to the left; but the point is that the change in the height,*0531

*"height change"--I will change that formally--height change slows down away from the y-axis.*0536

*But as we get close to the y-axis, as we get closer and closer, height will "blow out."*0545

*The height is going to "blow out": near 0, the function "blows out," and by that I mean that it blows out to either positive or negative infinity.*0551

*And as we approach 0 from the right side, we go out to positive infinity; as we approach from the left side, we go out to negative infinity.*0568

*However, there is one point that simply isn't allowed: 0 is not in domain--why? because 1/0 would be not allowed.*0578

*We are not allowed to take dividing by 0; so 0 is not in domain.*0587

*We can talk about what happens to 0.00000001, but we can't talk about what happens to 0 itself.*0591

*So, 0 is not in the domain; but near 0, as we approach 0, it goes out to either positive or negative infinity.*0599

*We will talk about this more when we talk about vertical asymptotes.*0605

*All right, that is spelled kind of weirdly, just in case you are curious to look at it right now: asymptotes--a weird spelling, but pronounced "aa-sim-tohts."*0610

*All right, the next one is the absolute value function.*0618

*For this one, it is only going to output positives; why? because absolute value only gives out positive numbers.*0621

*If you put in -3, it becomes positive 3; if you put in positive 3, it becomes positive 3.*0633

*Whatever you put in, it is stripped of negative numbers; it has to come out as a positive number.*0637

*Notice that it is also kind of similar to f(x) = x: when we have that normal f(x) = x, it would keep going like this.*0642

*That is the thing to notice--that in a way, it flips its direction upon touching the x-axis.*0655

*It flips direction when touching..."upon touching"--I will make it exactly correct...upon touching the x-axis; it flips the direction that it is going in.*0665

*What I mean by that: let's imagine that we start over here at -10; we would get positive 10.*0681

*We are moving this way; we now plug in -8; we get positive 8; we plug in -4; we get positive 4.*0687

*We are going this way; we are going this way; we are going this way; we are going this way.*0691

*All of a sudden, we hit a height of 0, and it bounces up; it flips to going this way right here.*0694

*And this is because it only outputs positives; so when we would get below the x-axis, it has to bounce off,*0702

*because otherwise we would be outputting a negative.*0708

*So, it flips the direction that it is going in upon touching the x-axis.*0710

*All right, now the trigonometric functions: these ones...there is a good chance you haven't seen these.*0715

*Or if you have, they are pretty new to you at this point.*0720

*The trigonometric functions--you will learn a lot about these in trigonometry.*0723

*But right now, the main thing I want to point out is the fact that they repeat.*0726

*Sine of x just does the same thing: see, this interval here is the same as this interval here, is the same as this chunk here, is the same as this chunk here.*0732

*We are just seeing it repeat: cosine of x, this chunk to this chunk, is the same as this chunk to this chunk; we are just seeing it repeat.*0742

*They are slightly different in how they are set up, but they are repeating functions.*0748

*Let's also look at it in different axes, so that you can understand what is going on better.*0752

*This is the classic -10 to 10 axes that we did for everything else; but how about some other ones?*0755

*It turns out that it does its variance between 0 and 2π; don't worry about actually understanding what is going on here.*0761

*I just want to have you see this stuff, so it is not totally new when you see it later.*0767

*You will get it all very well in trigonometry.*0772

*From 0 to 2π, we have one repetition; from -2π to 0, we have another repetition.*0775

*It repeats itself every 2π; it also varies between 1 and -1, both for sine and cosine.*0782

*It varies between 1 and -1, and it also has repetitions on a 2π basis, both for cosine and sine.*0790

*We will see why this is the case when we actually study trigonometry.*0797

*But the main thing to get out of this right now is that trigonometric functions are these repeating functions,*0800

*that they are a way of being able to see the same thing happen after we go down far enough.*0804

*We go down a certain amount, and it becomes the same thing; go down a certain amount, and it becomes the same thing; they repeat themselves.*0808

* Exponential and logarithmic functions: all right, we are back to our -10 to 10, our standard axes that we were used to before.*0816

*The thing to notice here: on exponential, it blows out really fast; look at how fast this manages to go out of our axis windows.*0822

*We get outside of being able to see this out of our viewing window so quickly.*0835

*By the time we have made it to 1, we are at 10; at 2, we are at 100; at 3, we are at 1,000.*0839

*So, exponential functions (10 ^{x} is only one possible exponential function)--they are going to go out really, really quickly.*0845

*They are going to just shoot up, having absolutely massive height.*0851

*Blowing out probably isn't the perfect word, since we used "blowing out" for asymptotes.*0854

*Let's instead say its height grows really fast.*0859

*Over here at log _{10}(x), look at how long it takes to even get up to 1; it takes us out to 10 to get it.*0865

*We can see that the height is slowing down; its increase in height slows down.*0870

*Its height increase slows massively; it grows really, really slowly.*0874

*That is probably the main important thing to get out of these.*0885

*Don't worry about actually understanding what is going on precisely right now.*0888

*We will have an entire section on this, when we talk about exponential and logarithmic functions in detail for an entire section.*0890

*But right now, I just want you to say, "Oh, exponential functions get really big really fast,*0896

*and logarithmic functions stay pretty low for a very long time."*0901

*Just to make a point of how long these sorts of things are, how slow and how fast they are,*0905

*for logarithmic and exponential, respectively, let's look at it with new axes.*0912

*So, for our exponential, we are going from only -3 to 3; by the time we have made it up to 3, we have hit 1,000.*0917

*1,000 is how big we have managed to get.*0926

*And it doesn't actually get to 0; it just looks like that--it is approaching it, because 10 ^{-3} would be 1/10^{3}, 1/1000.*0929

*So, it is just really close to the x-axis.*0937

*Once again, don't worry about understanding this perfectly right now; we will talk about it later.*0939

*Logarithmic functions: they are going to take forever to even get to reasonable numbers.*0942

*We have to get to 1,000 before we even manage to make it up to a height of 3.*0948

*So, they grow really, really slowly; the height growth on logarithmic functions is really, really slow,*0954

*and slows down massively the farther you get out; whereas exponential functions are really, really fast,*0959

*and increase massively the farther you get out.*0964

*All right, of course, the functions we see--that makes it for our petting zoo--we have completed our petting zoo;*0967

*but when we see functions out in the wild, they are normally not going to end up being in their pure form.*0973

*They are not going to be x ^{2} or √x.*0979

*Normally, they have had other things put on them or added on them; they have been shifted, stretched, or flipped.*0982

*They have been transformed in some way.*0987

*Still, it helps to know the general shape for a function before transformation.*0990

*Beyond these shiftings, stretchings, and flippings, we can still have a pretty good idea of what is going on.*0994

*Other times, functions will be mixed with other functions.*1001

*We might have things like f(x) = x ^{3} + x^{2}: that is not just one pure function--that is x^{3} and x^{2}.*1004

*Or x ^{2} times √x--once again, that is not just one pure function; that is two functions mixed together.*1010

*Or h(x) = |x ^{3}|: once again, that is not just one pure function; it is two things put together; it is absolute value and x^{3}.*1016

*Once again, though, it helps to know each function's general form before trying to figure out how they interact.*1024

*If we understand how |x| works, and we understand how x ^{3} works, it will make sense to us*1030

*when we work on the graph of the absolute value of x ^{3}.*1035

*We will have a better understanding of what is going on--what we are seeing.*1038

*We will learn about both of the above ideas in the lessons Transformations of Functions--the transformations*1042

*where we shift, stretch, and flip will be in the Transformations of Functions lesson; and we will talk*1047

*about composite functions--we will talk about arithmetic combinations (these first two are arithmetic combinations--*1052

*once again, don't worry if you don't know what these things mean precisely; we have lessons for that);*1058

*and then finally, an actual composite function, where we combine the way that two functions are working.*1062

*We will learn about these in much greater detail in those two respective sections.*1066

*All right, great; we are ready for some examples.*1070

*Here are two graphs without axes; they are the graphs of what functions?*1074

*Well, this one is going up; it looks like a slope of 1; it is pretty stable; it is just increasing continuously.*1077

*It doesn't bounce; so this is almost certainly f(x) = x; great.*1083

*And this one: one side goes up; we see this sort of blowing out--it is approaching -∞ (that is what it is going down to here);*1089

*and it is going up to positive ∞ up here; so what blew out and what got really, really slow in its height change?*1098

*Oh, yes, it is the reciprocal function: f(x) = 1/x; great.*1107

*Next, here are two functions that have been shifted, stretched, and/or flipped; what are the base functions making them up?*1115

*For this first one, the red graph, we think, "Oh, well, it looks kind of like a parabolic arc, but a parabolic arc on its side."*1120

*Oh, look over here: there is nothing over on the left side.*1128

*So, if there is nothing on the left side, it has cut out the negative side; it has cut out the left side.*1134

*What cut out one side? √x--we have that √x, and its height increase is still slowing.*1139

*Height increase slows the farther it gets out; so that is both of the identifying marks of being a √x function.*1147

*It is very different than the normal √x; but the basic function that is making this up is √x.*1155

*It has been shifted; it has been stretched; but it is still √x.*1161

*What about this one over here? Well, it is going down on both sides.*1166

*And its height increase gets faster...well, "height increase" is incorrect; it is not height increase, because we are going down.*1174

*But it is a height change, and that is the more fundamental idea about x ^{2}.*1187

*It isn't necessarily that it has to be going up, but that the change, the rate that it is going up*1193

*or the rate that it is going down, possibly, is continuing to increase.*1197

*Height change speeds up the farther we get out from the center; the center has been moved in this one.*1201

*The height change will speed up; we get faster and faster changes in our height.*1207

*So once again, this is f(x) = x ^{2}: it has been shifted, it has been flipped, and it has been stretched;*1211

*but we can still recognize that that is a parabola; it must be, at heart, coming from that same idea as behind x ^{2}; great.*1217

*Example 3: we want to graph f(x) = x ^{3}, g(x) = x^{2}, and h(x) = x, all on the same axes.*1225

*And we also want to set up the axes such that we go from -10 to 10 on the horizontal,*1233

*and none of the graphs are cut off vertically; so that means we can't lose any vertical information.*1239

*To help us understand what is going on here, let's make a table.*1246

*Now, we know what x ^{3} looks like, what x^{2} looks like, and what x looks like, in general.*1248

*So, we can use that information to help us out.*1253

*Let's see what the extreme values are and what the middle values are.*1255

*So, here is our table: x, f(x)...here is x at -10 and x at positive 10.*1258

*f(x)...let's make all three of them: x ^{3}, x^{2}, and x.*1269

*Let's actually use colors for these: x ^{2} will be in red, and blue will be x; great.*1278

*So, if we plug in -10, we are going to get -10 cubed, which is -1,000.*1285

*x ^{2} will become 100, positive, because the negatives cancel out; and x will become -10.*1292

*If we go to the other extreme at 10, that will be at positive 1,000; x ^{2} will also be at positive 100, and x will be at positive 10.*1302

*So, let's try some other things: let's look at what happens in the middle.*1313

*Well, in the middle, x ^{3} is at 0; x^{2} is at 0; and x is also at 0.*1315

*Let's see what happens in the middle between the middles: if we plug in, say, -5 and 5:*1322

*-5 times -5 is positive 25, times another -5...we get -125.*1330

*Over here, we will have positive 125; the red...in this, we will get squaring, so we will be at positive 25, positive 25.*1337

*And we will have -5 and positive 5.*1346

*And finally, at 1 and -1, -1 will go to -1; positive 1 will go to positive 1 for x ^{3}.*1349

*For x ^{2}, -1 will get cancelled to a positive; a positive is still for the positive; and then, -1 and positive 1.*1357

*The thing to notice here is that, when it is close to 0, they are not that different.*1365

*But the farther we get from 0, the more their differences become apparent.*1369

*So, let's make our graph; we will do it in blue, which hopefully won't be too confusing,*1373

*even though blue is connected to x; so what is the maximum vertical height that we have to have?*1382

*The maximum vertical height that we have to have is a huge 1,000, because we get up to -1,000 and positive 1,000.*1388

*So, we have to be at -1,000 and positive 1,000 as the vertical extremes.*1395

*Here is the middle at -500, positive 500; and then, we will actually go to an extreme*1400

*of 10 and -10 horizontally, because that is what we were told to do.*1410

*And here is our middle at 5, and middle at -5; and here is 1 and positive 1; great.*1413

*All right, so at this point, let's graph x ^{3}; this is probably going to be one of the easiest to graph.*1422

*At 0, it is at 0; at 1, it manages to be at 1, so it has barely even gotten off the x-axis.*1426

*At 5, it is at 125, so it is a little bit over 1/5 of the way to the 500; so let's say it is around there.*1431

*And then, 10 is going to be all the way up at 1,000.*1437

*-1 to -1 is barely off of the x-axis; -5 will be at -125, so we are little past, but close to...probably a little too far down, actually...*1440

*we are a little over 1/5 of the way to the 500; and then finally, at -10, we are all the way at a huge -1,000.*1451

*Its curve is going to look like this; it manages to grow massively very, very quickly, as it gets farther and farther away from the y-axis--*1459

*as it gets farther and farther from the center of its graph.*1476

*What about x ^{2}? We get 1 and 1 in the same location; at 5, it is at a 25--a meager, tiny jump above;*1479

*and then, at 10, it is at 100, so it is a little bit below the height of x ^{3} at 5.*1487

*The same on the reverse side: the parabola...when we look at it this far out, it is growing fast;*1493

*it manages to get to 100 by the time it has gotten to 10; but it is still tiny, tiny;*1508

*it looks so stout---it looks so short--compared to x ^{3}.*1515

*Finally, we look at what happens to x, just the plain identity function.*1519

*And at 10, it manages to only be 10 above; we are talking about there; at -10, it is only -10.*1523

*So, we have it barely, barely growing off of that x-axis; it is barely breaking away, when compared to these giants like x ^{2} and x^{3}.*1529

*So, when we are really, really close to actually being near 0, when we are really close to the center of these graphs, they are very similar.*1543

*But when we look at them on a larger scale, not even that big--just -10 to 10--suddenly the differences become apparent.*1551

*They become massive, huge differences: the difference between x ^{3} and x^{2} at just 10 out is a difference of 900.*1556

*There are huge differences between these; and they get even bigger, the farther we go out.*1563

*The final example, Example 4: Think about functions of the form f(x) = x ^{n}, where n is a positive integer--it is contained in the natural numbers.*1568

*Then distinguish the difference between when n is odd and when n is even.*1578

*Let's look at some examples for when n is odd: n is odd would be like x, or maybe x ^{3} or x^{5}.*1582

*So, we know what x looks like; it is just like that.*1591

*x ^{3} blows out pretty quickly; x^{5} blows out even faster.*1595

*By the time it makes it to 2, it is at 2, 4, 8, 16, 32 height.*1602

*So, by the time x ^{5} has an input of 2, it is getting an output of 32; so it blows out really, really fast.*1608

*Let's compare some even ones; say we have x ^{2}...well, we know what that one looks like; it just looks like a parabola.*1618

*x ^{4}...well, it is like a parabola, but it grows even faster.*1625

*By the time we get to positive 2, instead of just being at 4, we are at 16.*1631

*So, it grows really fast--not quite as fast as x ^{5}, but faster than x^{3}.*1635

*And as we go to the other side, since (-2) ^{2} is positive 4, (-2)^{4} is positive 4 times positive 4, or positive 16.*1640

*So, as long as we are even, we are going to cancel out those negatives.*1650

*And that is the idea that we see right here: look, when we are even, we go off in the same direction.*1654

*If we were to do this for x ^{6}, it would be the same thing, but growing even faster.*1659

*We would just be growing even faster; so what we are seeing here is that, when n is odd,*1663

*that means that the ends of the graph go in opposite directions.*1673

*So, the ends of the graph go in opposite directions if n is odd.*1679

*But if n is even, the ends go in the same direction.*1686

*That is the major difference between these; in many ways, they are very similar--the higher the n, the faster we have this growth rate.*1696

*But depending on if we are odd or if we are even, that changes whether or not the two ends will point in the same direction.*1706

*If it is even, they are both pointing up, as long as there isn't a negative in front.*1711

*And if it is odd, one of them is going to be pointing down; the first one is going to be pointing down,*1715

*as long as there is not a negative in front, because a negative raised to an odd number remains a negative.*1718

*So, if n is odd, we have opposite directions; if n is even, we have the same direction.*1724

*All right, great; that finishes up for the examples; I hope you have a good idea of the various functions out there.*1734

*There are a lot of functions out there; but at this point, you probably have a reasonable understanding of what they are doing.*1738

*And the more you continue to do math, functions are just going to make more and more sense.*1742

*Just pay attention to what you are doing, and say, "Oh, yes, I have seen this one before,"*1746

*or "Oh, I haven't seen this one before"--pay attention to what it looks like, and then,*1750

*the next time you see something like that one, you will be able to apply that information*1753

*and have a better idea of how to draw that curve.*1757

*All right, I will see you at Educator.com later--goodbye!*1759

1 answer

Last reply by: Professor Selhorst-Jones

Fri Apr 17, 2015 4:22 PM

Post by Micheal Bingham on April 17, 2015

During your fourth example, you began listing x^n with n=2 for the even "n" functions. What would happen if you compared even lower, yet still even (or odd) values of n. For example, the graphs of: x^-2 or x^0 (constant function) and/or x^-5 do not seem easily comparable to the other positive integer (n) functions. Is there still a way in which these negative numbered exponent graphs relate to their positive integer ones (with even/odd)?

1 answer

Last reply by: Professor Selhorst-Jones

Sun Mar 9, 2014 3:37 PM

Post by Linda Volti on February 24, 2014

For the square root function, why can't an output be negative? This would mean there are two outputs for one input and that, therefore, it would not be a function. Isn't the square root of, say, 4 +/- 2?

1 answer

Last reply by: Professor Selhorst-Jones

Thu Jul 11, 2013 12:17 PM

Post by Jonathan Traynor on July 1, 2013

Outstanding!!!!!!

3 answers

Last reply by: Professor Selhorst-Jones

Tue Jun 4, 2013 11:26 PM

Post by Rajendran Rajaram on June 4, 2013

hello,

I have an important question does a cube and a rectangular prism that has equal surface area have equal volumes.

thank you