For more information, please see full course syllabus of Math Analysis

For more information, please see full course syllabus of Math Analysis

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### Properties of Functions

- Over an interval of x-values, a given function can be
*increasing*,*decreasing*, or*constant*. That is, always__going up__, always__going down__, or__not changing__, respectively. This idea is easiest to understand visually, so look at a graph to find where these things occur. - We talk about
*increasing*,*decreasing*, and*constant*in terms of__intervals__: that is, sections of the horizontal axis. Whenever you talk about one of the above as an interval, you always give it in__parentheses__. - An (
*absolute*/*global*)*maximum*is where a function achieves its__highest__value. An (*absolute*/*global*)*minimum*is where a function achieves its__lowest__value. - A
*relative maximum*(or*local maximum*) is where a function achieves its highest value in some "neighborhood". A*relative minimum*(or*local minimum*) is where a function achieves its lowest value in some "neighborhood". [Notice that these aren't necessarily the highest/lowest locations for the entire function (although they might be), just an extreme location in some interval.] - We can refer to all the maximums and minimums of a function (both absolute and relative) with the word
*extrema*: the extreme values of a function. - We can calculate the
*average rate of change*for a function between two locations x_{1}and x_{2}with the formulaf(x _{2}) − f(x_{1})x_{2}−x_{1}. - It is often very important to know what x values for a function cause it to output 0, that is to say, f(x) = 0. This idea is so important, it goes by many names: the
*zeros*of a function, the*roots*, the*x-intercepts*. But these all mean the same thing:__all x such that f(x) = 0__. - An
*even*function (totally different from being an even number) is one where

In other words, plugging in the negative or positive version of a number gives the same output. Graphically, this means that even functions are symmetric around the y-axis (mirror left-right).f(−x) = f(x). - An
*odd*function (totally different from being an odd number) is one where

In other words, plugging in the negative version of a number gives the same thing as the positive number did, but the output has an additional negative sign. Graphically, this means that odd functions are symmetric around the origin (mirror left-rightf(−x) = − f(x). __and__up-down).

### Properties of Functions

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
- Increasing Decreasing Constant
- Find Intervals by Looking at the Graph
- Intervals Show x-values; Write in Parentheses
- Maximum and Minimums
- Relative (Local) Max/Min
- Max/Min, More Terms
- Average Rate of Change
- Zeros/Roots/x-intercepts
- Even Functions
- Odd Functions
- Even/Odd Functions and Graphs
- Example 1
- Example 2
- Example 3
- Example 4

- Intro 0:00
- Introduction 0:05
- Increasing Decreasing Constant 0:43
- Looking at a Specific Graph
- Increasing Interval
- Constant Function
- Decreasing Interval
- Find Intervals by Looking at the Graph 5:32
- Intervals Show x-values; Write in Parentheses 6:39
- Maximum and Minimums 8:48
- Relative (Local) Max/Min 10:20
- Formal Definition of Relative Maximum
- Formal Definition of Relative Minimum
- Max/Min, More Terms 14:18
- Definition of Extrema
- Average Rate of Change 16:11
- Drawing a Line for the Average Rate
- Using the Slope of the Secant Line
- Slope in Function Notation
- Zeros/Roots/x-intercepts 19:45
- What Zeros in a Function Mean
- Even Functions 22:30
- Odd Functions 24:36
- Even/Odd Functions and Graphs 26:28
- Example of an Even Function
- Example of an Odd Function
- Example 1 29:35
- Example 2 33:07
- Example 3 40:32
- Example 4 42:34

### Math Analysis Online

### Transcription: Properties of Functions

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

*Today, we are going to talk about properties of functions.*0002

*Functions are extremely important to math; we keep talking about them, because we are going to use them a lot; they are really, really useful.*0005

*To help us investigate and describe behaviors of functions, we can talk about properties that a function has.*0012

*There are a wide variety of various properties that a function can or cannot have.*0017

*This lesson is going to go over some of the most important ones.*0022

*While there are many possible properties out there that we won't be talking about in this lesson,*0024

*this lesson is still going to give us a great start for being able to describe functions,*0028

*being able to talk about how they behave and how they work.*0032

*So, this is going to give us the foundation for being able to talk about other functions in a more rigorous way,*0035

*where we can describe exactly what they are doing and really understand what is going on; great.*0039

*All right, the first one: increasing/decreasing/constant: over an interval of x-values,*0045

*a given function can be increasing, decreasing, or constant--that is, always going up, always going down, or not changing.*0050

*Its number will always be increasing; its number will always be decreasing; or its number will be not changing.*0060

*And by number, I mean to say the output from the inputs as we move through those x-values.*0066

*This is really much easier to understand visually, so let's look at it that way.*0072

*So, let's consider a function whose graph is this one right here: this function is increasing on -3 to -1.*0076

*We have -3 to -1, because from -3 to -1, it is going up; but it stops right around here.*0083

*So, it stops increasing on -3; it stops increasing after -1, but from -3 to -1, we see that it is increasing.*0091

*It is probably increasing before -3, but since all we have been given is this specific viewing window to look through,*0099

*all we can be guaranteed of is that, from -3 to -1, it is increasing.*0104

*Then, it is constant on -1 to 1; it doesn't change as we go from -1 to 1--it stays the exact same, so it is constant on -1 to 1.*0108

*However, it is increasing before -1, and it is decreasing after 1; so it is decreasing on 1 to 3, because we are now going down.*0121

*So, it continues to go down from 1 on to 3, because we can only be guaranteed up until 3.*0131

*It might do something right after the edge of the viewing window, so we can only be sure of what is there.*0138

*It is decreasing from 1 to 3--great; that is what we are seeing visually.*0143

*It is either going up, straight, or down; it is either horizontal, it is going up, or it is down.*0147

*Increasing means going up; constant means flat; decreasing means down.*0153

*Formally, we say a function is increasing on an interval if for any a and b in the interval where a < b, then f(a) < f(b).*0160

*Now, that seems kind of confusing; so let's see it in a picture version.*0167

*Let's say we have an interval a to b, and this graph is above it.*0172

*If we have some interval--it is any interval, so let's just say we have some interval--that is what was between those two bars--*0179

*and then we decide to grab two random points: we choose here as a and here as b;*0188

*a is less than b, so that means a is always on the left side; a is on the left, because a is less than b.*0195

*That is not to say b over c; that is because--I will just rewrite that--we might get that confused in math.*0204

*a is on the left because a is less than b; if we then look at what they evaluate to, this is the height at f(a), and then this is the height at f(b).*0211

*And notice: f(b) is above f(a); f(b) is greater than f(a).*0224

*So, it is saying that any point on the left is going to end up being lower than points on the right, in the interval when it is increasing.*0231

*In other words, the graph is going up in the interval, as we read from left to right.*0240

*Remember, we always read graphs from left to right.*0245

*So, during the interval, we are going from left to right; we are going up as we go from left to right.*0248

*We have a similar thing for decreasing; we have some interval, some chunk, and we have some decreasing graph.*0255

*And if we pull up two points, a and b (a has to be on the left of b, because we have a < b), then f(a) > f(b).*0265

*f(a) > f(b); so decreasing means we are going down--the graph is going down from left to right.*0277

*We don't want to get too caught up in this formal idea; there is some interval, some place,*0287

*where if we were to pull out any two points, the one on the left will either be below the one on the right,*0293

*if it is increasing; or if it is decreasing, it will be above the one on the right.*0299

*We don't want to get too caught up in this; we want to think more in terms of going up and going down, in terms of reading from left to right.*0304

*Finally, constant: if we have some interval, then within that interval, our function is nice and flat,*0310

*because if we choose any a and b, they end up being at the exact same height.*0318

*There is no difference: f(a) = f(b); the graph's height does not change in that interval.*0324

*While the definitions on the previous slide give us formal definitions--they give us something that we can really understand*0332

*if we want to talk really analytically--we don't really need to talk analytically that often in this course.*0338

*It is going to be easiest to find these intervals by analyzing the graph of the function.*0343

*We just look at the graph and say, "Well, when is it going up? When is it going down? And when is it flat?"*0347

*That is how we will figure out our intervals.*0352

*We won't necessarily be able to find precise intervals; since we are looking at a graph, we might be off by a decimal place or two.*0354

*But mostly, we are going to be pretty close; so we can get a really good idea of what these things are--*0360

*what these intervals of increasing, decreasing, or constant are.*0366

*So, we get a pretty good approximation by looking at a graph.*0370

*And if you go on to study calculus, one of the things you will learn is how to find increasing, decreasing, and constant intervals precisely.*0373

*That is one of the major fields, one of the major uses of calculus.*0380

*You won't even need to look at a graph; you will be able to do it all from just knowing what the function is.*0383

*Knowing the function, you will be able to turn that into figuring out when it is increasing, when it is decreasing, and when it is flat.*0387

*You will even be able to know how fast it is increasing and how fast it is decreasing.*0392

*So, there is pretty cool stuff in calculus.*0395

*All right, intervals show x-values: for our intervals of increasing, decreasing, and constant,*0399

*remember, we are giving intervals in terms of the x-values; it is not, not the points.*0403

*We describe a function's behavior by saying how it acts within two horizontal locations.*0411

*We are saying between -5 and -3, horizontally; it is not the point (-5,-3); it is between the locations -5 and -3.*0416

*And don't forget, we always read from left to right; it is reading from left to right, as we read from -5 up until -3.*0425

*So, the other thing that we need to be able to do is: we need to always put it in parentheses.*0432

*Parentheses is how we always talk about increasing, decreasing, and constant intervals.*0437

*Why do we use parentheses instead of brackets?*0442

*Well, think about this: a bracket indicates that we are keeping that point;*0444

*a parenthesis indicates that we are dropping that point, not including that in the interval.*0449

*But the places where we change over, the very end of an interval, is where we are flipping*0453

*from either increasing to decreasing or increasing to constant; we are changing from one type of interval to another.*0457

*So, those end points are going to be changes; they are going to be places where we are changing from one type of interval to another.*0463

*So, we can't actually include them, because they are switchover points.*0470

*We want to only have the things that are actually doing what we are talking about.*0473

*The switchovers will be switching into something new; so we end up using parentheses.*0476

*All right, a really quick example: if we have f(x) = x ^{2} - 2x, that graph on the left,*0480

*then we see that it is decreasing until it bottoms out here; where does it bottom out?*0485

*It bottoms out at 1, the horizontal location 1; and it is decreasing all the way from negative infinity, out until it bottoms out at 1.*0489

*And then, it is increasing after that 1; it just keeps increasing forever and ever and ever.*0499

*So, it will continue to increase out until infinity.*0503

*So, parenthesis; -∞ to 1 decreasing; and increasing is (1,∞).*0506

*We don't actually include the 1, because it is a switchover.*0513

*At that very instant of the 1, what is it--is it increasing? Is it decreasing?*0515

*It is flat technically; but we can't really talk about that yet, until we talk about calculus.*0520

*So for now, we are just not going to talk about those switchovers.*0524

*All right, the next idea is maximums and minimums.*0527

*Sometimes we want to talk about the maximum or the minimum of a function, the place where a function attains its highest or lowest value.*0530

*We call c a maximum if, for all of the x (all of the possible x that can go into the function), f(x) ≤ f(c).*0538

*That is to say, when we plug in c, it is always going to be bigger than everything else that can come out of that function,*0545

*or at the very least equal to everything else that can come out of the function.*0553

*A minimum is the flip of that idea; a minimum is f(c) is going to be smaller or equal to everything else that can be coming out of that function.*0556

*So, a maximum is the highest location a function can attain, and a minimum is the lowest location a function can attain.*0565

*On this graph, the function achieves its maximum at x = -2; notice, it has no minimum.*0575

*So, if we go to -2 and we bring this up, look: the highest point it manages to hit is right here at -2.*0580

*Why does it not have a minimum? Well, if we were to say any point was its minimum--look, there is another point that goes below it.*0589

*So, since every point has some point that is even farther below it, there is no actual minimum,*0594

*because the minimum has to be lower than everything else.*0600

*There is a maximum, because from this height of 3, we never manage to get any higher than 3, so we have achieved a maximum.*0603

*And that occurs at x = -2; great.*0610

*We can also talk about something else; first, let's consider this graph,*0615

*this monster of a function, -x ^{4} + 2x^{3} + 5x^{2} - 5x.*0619

*Technically, this function only has one maximum; you can only have one maximum,*0624

*and it is going to be here, because it is the highest point it manages to achieve; it would be x = 2.*0630

*But it actually has no minimum; why does it not have any minimum?*0636

*Well, it kind of looks like this is the low point; but over here, we managed to get even lower.*0638

*Over here, we managed to get even lower; and because it is just going to keep dropping off to the sides,*0643

*forever and ever and ever, we are going to end up having no minimums whatsoever in this function,*0647

*because it can always go lower; there is no lowest point it hits; it always keeps digging farther down.*0651

*But nonetheless, even though there is technically only one maximum and no minimums at all,*0659

*we can look at this and say, "Well, yes...but even if that is true, that there isn't anything else,*0663

*this point here is kind of interesting; and this point here is kind of interesting,*0669

*in that they are high locations and low locations for that area."*0674

*This is the idea of the relative minimum and maximum; we call such places--these places--*0679

*the highest or lowest location (I will switch colors...blue...oh my, with yellow, it has managed...blue here; green here)...*0686

*relative minimums are the ones in green, and the relative maximums are the ones in blue.*0701

*And sometimes the word "local" is also used instead; so you might hear somebody flip between relative or local, or local or relative.*0708

*These places are not necessarily a maximum or a minimum for the entire function, for every single place.*0715

*But they are such a maximum or minimum in their neighborhood; there is some little place around them where they are "king of their hill."*0721

*So, this one is the maximum in this interval, and this one is the minimum in this interval; and this one is the maximum in this interval.*0730

*But if we were to look at a different interval, there would be no maximum or minimum in this interval,*0737

*because it just keeps going down and down and down.*0740

*And if we were to look at even in here, it is clearly right next to them--if we were to put a neighborhood around this, it would keep going down.*0743

*It is not the shortest one around; it is not the highest one around.*0749

*But these places are the highest or lowest in their place.*0754

*OK, so this gives us the idea of a relative maximum or a relative minimum.*0761

*Formally, a location, c, on the x-axis is a relative maximum if there is some interval,*0765

*some little place around that, some ball around that, that will contain c, such that,*0770

*for all x in that interval, f(x) ≤ f(c)--in its neighborhood, c is the highest thing around.*0775

*It is greater than all of the other ones.*0783

*Similarly, for a relative minimum, there is some interval such that f(x) is going to be less than or equal to f(c).*0785

*In its neighborhood, it is the lowest one around; lowest one around makes you a minimum--highest one around makes you a maximum--*0792

*that is to say, a relative maximum or a relative minimum.*0799

*Once again, this is sort of like what we talked about before with the previous formal definition for maximum and minimum,*0804

*and also for the formal definition of intervals of increasing, decreasing, and constant.*0809

*Don't get too caught up on what this definition means precisely.*0813

*The important thing is that we have this graphical picture in our mind that relative maximum just means the high point in that area.*0817

*And relative minimum just means the low point in that area; that is enough for us to really understand what is going on here.*0823

*Getting caught up in these precise things is really something for a late, high-level college course to really get worried about.*0829

*For now, it is enough to just get an idea of "it is the high place" or "it is the low place."*0835

*Don't forget: the terms relative and local mean basically the same thing--actually, they mean exactly the same thing.*0841

*They can be used totally interchangeably; and some people prefer to use one; some people prefer to use another.*0846

*Some people will flip between the two; so don't get confused if you hear one or you hear another one; they just mean the same thing.*0851

*To distinguish relative local maximums and minimums from a maximum and minimum over the entire function,*0859

*we can use the terms "absolute" or "global" to denote the latter.*0864

*If we want to say it is the maximum over the entire function, we could call it the absolute maximum or the global maximum.*0868

*So, an absolute, global maximum/minimum is where the function is highest/lowest over the entire function,*0875

*which is exactly how we defined maximum/minimum at first, before we started to talk about the idea of relative maximum/relative minimum.*0881

*So, absolute or global maximum/minimum is over the entire function--the function's highest/lowest over everywhere in the domain.*0886

*If we want to talk about all of the relative or absolute maximums/minimums in the functions, we can call the them the extrema (or the "ex-tray-ma").*0899

*Why? Because they are the function's extreme values: they are the extreme high points*0906

*and the extreme low points that the function manages to go through, so we can call them the extrema.*0911

*So, there we are; there is just something for us: extrema.*0916

*If we want to talk about relative or absolute maximums/minimums in general, we use this word to do it.*0919

*And absolute or global talks about the single highest or single lowest;*0924

*relative just talks about one that is high or low in its neighborhood, in the area around that point.*0928

*Just like find increasing/decreasing/constant intervals, we want to do this from the graph.*0936

*We don't want to really get too worried or too caught up on these very specific definitions,*0941

*the formal definitions we were talking about on the previous slide.*0945

*We just want to say, "OK, yes, we see that that is a high point on the graph; that is a low point on the graph."*0948

*So, find your minimums; find your maximums by looking at the graph.*0953

*And once again, if you go on to study calculus, you will learn how to find extrema precisely, without even needing to look at a graph.*0957

*You will be able to find them exactly; you won't have to be doing approximations because you are looking at a graph.*0963

*And you won't even have to look at a graph to find them.*0966

*So once again, calculus is pretty cool stuff.*0968

*Average rate of change: this also can be called average slope.*0972

*When we talked about slope in the introductory lessons, we discussed*0976

*how it can be interpreted as the rate of change, how fast up or down the line is moving.*0978

*If we have a line like this, it is not moving very fast up; but if we have another line like this, it is moving pretty quickly up.*0984

*So, it is a rate of change; the slope is how fast it is changing--the rate of change; how fast are we going up?*0993

*Now, most of the functions we are going to work with aren't lines; but we can still use this idea.*0999

*We can discuss a function's average rate of change between two points.*1003

*So, if an imaginary line is drawn between two points on a graph, its slope is the average rate of change.*1009

*Say we take two points, this point here and this point here; and we draw an imaginary line between them.*1015

*Then, the slope of that imaginary line is the average rate of change,*1022

*because what it took to get from this point to the second point is that we had to travel along this way.*1025

*And while we actually went through this curve here--we actually went through this curve,*1031

*but on the whole, what we managed to do, on average, is: we really just kind of went along on that line.*1037

*We could forget about everything we went through, and we could just ask, "Well, what is the average thing that happened between these two points?"*1045

*And that would be our average rate of change--how fast we were moving up from our first point to our second point.*1050

*So, if we want to find the average rate of change, how do we do this?*1057

*Let's say we have two locations, x _{1} and x_{2},*1061

*and we want to find the slope of that imaginary line between those two points on the function graph.*1065

*So, that line is sometimes called the secant line; for the most part, you probably won't hear that word too often.*1072

*But in case it comes up, you know it now.*1078

*Remember, if we want to find what the slope of this imaginary line is, the slope of this secant line, we know what slope is.*1080

*How do we find slope? Remember, slope is the rise over the run, so it is the difference between our heights*1088

*y _{2} and y_{1}, our second height and our first height--what did our height change by,*1094

*and what did our horizontal location change by--our second location minus our first location?*1099

*So, our horizontal distance is x _{2} - x_{1}; and our vertical distance is y_{2} - y_{1}.*1104

*So, y _{1} is the height over here; y_{2} is the height over here.*1115

*y _{2} - y_{1} over x_{2} - x_{1} is the rise, divided by the run.*1119

*But what are y _{1} and y_{2}?--if we want to figure out what y_{1} and y_{2} are,*1127

*well, we just need to look at what x _{1} and x_{2} are.*1132

*So, since x _{1} and x_{2} are coming to get placed by the function,*1134

*then y _{2}'s height is really just f(x_{2}), because that is how the graph gets built.*1141

*The input gets dropped to an output; we map an input to an output.*1146

*And y _{1} over here is from f(x_{1}).*1151

*So, since our original slope formula is y _{2} - y_{1} over x_{2} - x_{1},*1156

*and we know that y _{1} is the same thing as f(x_{1}) and y_{2} is the same thing as f(x_{2}),*1160

*we can just plug those in, and we get the change in our function outputs, f(x _{2}) - f(x_{1}),*1164

*divided by our horizontal distance, x _{2} - x_{1}.*1171

*For our average rate of change, we just look at how much our function changed by between those horizontal locations.*1174

*How much did its output change by? Divide that by how much our distance changed by.*1180

*It is often really useful and important to find what inputs cause a function to output 0.*1186

*So, if we have some function f, we might want to know what we can put into f that will give out 0.*1191

*That is the values of x such that f(x) = 0.*1197

*Graphically, since f(x)...remember, f(x) is always the vertical component; the outputs come to the vertical;*1200

*so, if our outputs are coming from the vertical, then 0 is going to be the x-axis.*1208

*We have a height of 0 here; so graphically, we see that this is where the function crosses the x-axis.*1213

*Our crossing of the x-axis is where f(x) = 0.*1219

*This idea of f(x) = 0 is so important that it is going to go by a bunch of different names.*1224

*It can be called the zeroes of a function; it can be called the roots of a function; and it can be called the x-intercepts.*1229

*x-intercepts--that makes sense, because it is where it crosses the x-axis.*1236

*Zeroes make sense, because it is where we have the zeroes showing up.*1241

*But how can we remember roots--why is roots coming out?*1245

*Well, one way to think about it--and actually where this word's origin is coming from--*1249

*is because it is the roots that the function is planted in.*1254

*The function we can think of as being planted in the ground (not literally the ground, but we can think of it as being the ground of the x-axis).*1258

*So, it is like the function has put down roots in the soil.*1265

*It is not exactly perfect, but that is one good mnemonic to help us remember.*1269

*"Roots" means where we are stuck in the soil; it is where we are stuck in the x-axis;*1273

*it is where we have f(x) equal to 0, or where we have an equation equal to 0.*1277

*But all of these things--zeroes, roots, x-intercepts--they all mean the same thing.*1282

*They are the x such that f(x) = 0; we can also use these for equations--*1287

*we might hear it as the zeroes of an equation, the roots of an equation, or the x-intercepts of an equation.*1291

*There is no one way to find zeroes for all functions.*1297

*We are going to learn, for some functions, foolproof formulas to find zeroes, to tell us if there are zeroes and what those zeroes are.*1301

*But for other functions, it can be very difficult--very, very difficult, in fact--to find the zeroes.*1308

*And although we are going to learn some techniques to help us on the harder ones, there are some that we won't even see*1312

*in this course, because they are so hard to figure out.*1316

*But right now, the important thing isn't being able to find them, but just knowing*1319

*that, when we say zeroes, roots, x-intercepts of a function, or an equation, we are just talking about where f(x) = 0.*1323

*So, don't get too caught up right now in being able to figure out how to get those x-values such that f(x) = 0.*1331

*Just really focus on the fact that when we say zeroes, roots, or x-intercepts, all of these equivalent terms,*1337

*we are just saying where the function is equal to 0--what are the places that will output 0?*1344

*Even functions: this is a slightly odd idea (that was an accidental joke).*1351

*Even functions: some functions behave the same whether you look left or right of the y-axis.*1357

*For example, let's consider f(x) = x ^{2}: it is symmetric around the y-axis.*1363

*What do I mean by this? Well, if we plug in f(-3), that is going to end up being (-3) ^{2}, so we get 9.*1368

*But we could also plug in the opposite version to -3; if we flip to the positive side, -3 flips to positive 3.*1375

*If we plugged in positive 3, then f(3) is 3 ^{2}, so we get 9, as well.*1383

*It turns out that plugging in the negative version of a number or the positive version of a number,*1390

*-3 or 3, we end up getting the same thing; for -2 and 2, we end up getting the same thing.*1395

*For -47 and 47, we end up getting the same thing.*1400

*So, whatever we plug in, as long as they are exact opposites horizontally--*1405

*they are the same distance from the y-axis--the points are symmetric around the y-axis--*1409

*they are going to come out to the same height; they are going to have the same output.*1414

*We call this property even; and I want to point out that it is totally different from being an even number.*1419

*It is different from an even number--not the same thing as that.*1423

*But we call this property even for a function.*1427

*A function is even if all of the x for its domain, for any x that we plug in...*1432

*if we plug in the negative x, that is the same thing as the positive x.*1437

*Plugging in f(-x) is equal to plugging in f(x); so we plug in -x into the function, and we get the same thing as if we had plugged in positive x.*1442

*We can flip the signs, and it won't matter, as long as it is just negative versus positive.*1451

*Why do we call it even? It has something to do with the fact that all polynomials where all of the exponents*1457

*end up being even exponents--they end up exhibiting this property.*1462

*But then, this property can be used on other things; so don't worry too much about where the name is coming from.*1465

*But just know what the property is: f(-x) = f(x).*1470

*Odd functions are the reverse of this idea: other functions will behave in the exact reverse.*1476

*The left side is the exact opposite of the right side; for example, f(x) = x ^{3} behaves like this.*1481

*If we plug in -3, we get -3 cubed, so we get -27; but if we had plugged in positive 3, we would get positive 3 cubed, so we would get positive 27.*1488

*So, you see, you plug in the negative version of a number, and you plug in the positive version of a number;*1500

*and you are going to get totally opposite answers.*1508

*However, they are only flipped by sign; -27 and 27 are still somewhat related.*1510

*They are very different from one another--they are opposites, in a way; but we can also think of them as being perfect opposites.*1515

*-27's opposite is positive 27; so an odd function is one that behaves like this everywhere.*1520

*We call this property odd; it is totally different, once again, from being an odd number.*1528

*A function is odd if, for all x in its domain, f(-x) is equal to -f(x).*1532

*And that is a little confusing to read; but what that means is that, if we plug in -x,*1538

*then that is going to give us the negative version of if we had plugged in positive x.*1543

*So, if we plug in a negative number, and then we plug in a positive number, the outputs*1549

*that come out of them will be positive-negative opposites.*1555

*One of them will be positive; the other one will be negative.*1559

*So, negative on one side and positive on one side means that the outputs will also be negative on one side and positive on the other side.*1561

*It is not necessarily going to be the case that the negative side will always put out negative outputs.*1568

*But it will be the case that it will be flipped if it is odd.*1572

*This will make a little more sense when we look at some examples.*1575

*And once again, why are we calling it odd?*1577

*Once again, don't worry too much about it, but it because it is connected to polynomials where all of the exponents are odd numbers.*1579

*But don't really worry about it; just know what the property is.*1585

*Even/odd functions and graphs: we can see these properties in the graphs of functions.*1589

*An even function is symmetric around the y-axis: it mirrors left/right, because when we plug in a positive number,*1593

*and we plug in a negative number, as long as they are the same number, they end up getting put to the same location.*1600

*They get output to the same place.*1605

*An odd function, on the other hand, is symmetrical around the origin, which means we mirror left/right and up/down,*1607

*because when we plug in the positive version of a number, it gets flipped to the negative side, but also shows up on the opposite side.*1613

*It flips to the negative height or the positive height; it flips the positive/negative in terms of height.*1622

*So, let's look at some examples visually; that will help clear this up.*1627

*An even one: f(-x) = f(x); let's see how this shows up; if we plug in 0.5, we get here; if we plug in -0.5, we get here.*1630

*And look, beyond the fact that I am not perfect at drawing, they came out to be the same height.*1642

*If we plug in 2.0, and we plug in -2.0, they came out to be the same height.*1648

*You plug in the negative number and the positive number, and they end up coming out to be the same height.*1661

*That is what it means to be even; and since all of the positives will be the same as the negatives,*1666

*we end up getting this nice symmetry across the y-axis; it is just a perfect flip.*1670

*If we took the two halves and folded them up onto each other, they would be exact perfect matches; it is just mirroring the two sides.*1676

*Odd is sort of the reverse of this: f(-x) = -f(x).*1684

*For example, let's plug in -1: we plug in -1, and it ends up being at this height, just a little under 2.*1690

*Let's see what happens when we plug in positive 1; when we plug in positive 1, it ends up being just a little under -2.*1699

*So, we flip the horizontal location; that causes our vertical location to flip.*1708

*Let's try another one: we plug in 2.0, and we are practically past it; so we should be just a little bit before 2.0.*1713

*And we plug in -2.0, once again, just a little past it; so we are just a little before -2.0.*1722

*And look: we end up being at the same distance from the x-axis, but in totally opposite directions.*1727

*2.0, positive 2.0, causes us to go to positive 4 in terms of height; but -2.0 causes us to go to -4 in terms of height.*1735

*So, they are going to flip; if you flip horizontally, you also flip vertically; and that is why we mirror left/right and mirror up/down.*1744

*We are not just flipping around the y-axis; we are flipping around the origin,*1752

*because we are flipping the right/left and the up/down; flipping around the origin is flipping the right/left and the up/down.*1759

*We mirror left/right; we mirror up/down; that is what is happening with an odd function.*1771

*All right, we are finally ready for some examples.*1776

*There are a bunch of different properties that we covered; now, let's see them in use.*1778

*The first example: Using this graph, estimate the intervals where f is increasing and decreasing.*1781

*Find the locations of any extrema/relative maximums/minimums.*1787

*And our function is -1.5x ^{4} + x^{3} + 4x^{2} + 3.*1790

*Now, that is just so we can have an idea that that is what that function looks like.*1794

*But we are not really going to use this thing right here; it is not really going to be that helpful for us figuring it out.*1798

*So first, let's figure out intervals where f is increasing or decreasing.*1802

*First, it is increasing from all the way down (and it sounds like we can probably trust*1807

*that it is going to keep going down, because we have -1.5x ^{4}); it is increasing up until...it looks like just after -1.0.*1812

*It is increasing from negative infinity (because it is going all the way to the left--it is going up*1824

*as long as we are coming from negative infinity, because it goes down as we go to the left, but we read from left to right),*1835

*so it is increasing from negative infinity up until...let's say that is -0.9, because it is just after -1.0.*1839

*And then, it is also going to be increasing from here...let's say it starts there...up until about this point.*1846

*So, where is that? It is probably about 1.4; so it is increasing from 0 up until 1.4.*1852

*Where is it decreasing? It is decreasing from this point until this point.*1864

*That was -0.9 that we said before; so we will go from -0.9 up until 0.*1875

*And then, it increased up until 1.4; so now it is going to be decreasing from 1.4.*1881

*And it looks like it is going to just keep going down forever, and it does indeed.*1887

*So, it is going to be all the way out until infinity; it is going to continue decreasing; great.*1890

*Now, let's take a look at the extrema; where are the relative maximums/minimums?*1895

*We have relative maximums/minimums at all of these flipovers that we have talked about, here, here, and here.*1902

*So, our relative maximum/minimum, our high location, the absolute maximum/minimum, is going to be up here.*1909

*Relative maximums: we have x =...we said that was 1.4, and that point is going to be 1.4.*1916

*Let's take a look, according to this...and it looks like it is just a little bit under 8; let's say 7.9.*1931

*And then, the other one, the lesser of them, but still a relative maximum--it is occurring at x = -0.9.*1940

*So, its point would be -0.9; and we look on the graph, and it looks like it is somewhere between 4 and 5.*1949

*It looks a hair closer to 5, so let's say 4.6; great.*1958

*Relative minimum--our low place: well, we can be absolutely sure of what the x is there--it is pretty clear that that is x = 0.*1962

*And what is the height that it is at right there? It looks like it is exactly on top of the 3, so it is (0,3).*1972

*We have all of the intervals of increasing and decreasing.*1978

*And we also have all of our extrema, all of our relative maximums and minimums; great.*1981

*Example 2: A ball is thrown up in the air, and its position in meters is described by location of t.*1987

*Distance of t is equal to -4.9t ^{2} + 10t, where t is in seconds.*1993

*OK, so we have some function that describes the height of the ball--where the ball is.*1998

*What is the ball's average velocity (speed) between 0 seconds and 1 second,*2004

*between 0 and 0.01 seconds, and between 0 and 2.041 seconds?*2007

*OK, at first, we have some idea...if we were to figure out what this function looks like, it is a parabola.*2012

*It has a negative here, so it is ultimately going to go down.*2017

*And it has the 10t here; if we were to graph it, it would look something like this.*2020

*And that makes a lot of sense, because if we throw a ball up, with time, the ball is going to go up and them come back down.*2024

*So, that seems pretty reasonable: a ball is thrown up in the air, and its position is given by this.*2031

*But how does speed connect to position? Well, we think, "What is the definition of speed?"*2036

*We don't exactly know what velocity is, necessarily; maybe we haven't taken a physics course.*2040

*But we probably know what speed is from before in various things.*2044

*Speed is distance divided by time, so distance over time equals speed.*2047

*It seems pretty reasonable that velocity is going to be the same thing.*2058

*That is not exactly true, if you have actually taken a physics course; but that is actually going to work on this problem.*2063

*We are going to have a good idea of what is going on with saying that that is true.*2068

*All right, so what is the ball's average velocity?*2072

*The average velocity is going to be the difference in its height, divided by the time that it took to make that difference in height.*2074

*So, we are going to be looking for distance.*2080

*If we have 2 times, time t _{1} to time t_{2}, it is going to be the location at time t_{2},*2084

*minus the location at time t _{1}, over the difference in the time, t_{2} - t_{1}.*2095

*Oh, and that makes a lot of sense; it is going to be connected, probably, to what we learned in this lesson,*2102

*since with student logic, they normally try to give us problems that are going to be based off of what we just learned.*2107

*So, t _{2} - t_{1}...this looks just like average rate of change.*2112

*The average rate of something's position--that would make sense, that how fast it is going is the rate of change; the thing is changing its location.*2116

*The rate at which you are changing your location is the velocity that you have; perfect.*2124

*Great; so we need to figure out what it is at 0 seconds and what it is at 1 second right away.*2128

*So, the location at 0 seconds; we plug that in...-4.9(0) ^{2}, plus 10(0)...that is just 0, which makes a lot of sense.*2133

*If we throw a ball up, at the very beginning it is going to be right at the height of the ground.*2140

*Distance at time 1 is going to be -4.9 times 1 ^{2}, plus 10 times 1; so we get 5.1.*2146

*If we want to figure out what is its average velocity between 0 seconds and 1 second, then we have d(1) - d(0)/(1 - 0), equals 5.1 - 0/1, which equals 5.1.*2159

*What are our units? Well, we had distance in meters, and time in seconds; so meters divided by seconds...we get meters per second.*2180

*That makes sense as a thing to measure velocity and speed.*2189

*All right, next let's look at between 0 and 0.1 seconds.*2192

*If we want to find 0.01 seconds, the location at 0.01 equals -4.9(0.01) ^{2} + 10(0.01).*2196

*Plug that into a calculator, and that is going to end up coming out to be 0.09951; so let's just round that up*2211

*to the much-more-reasonable-to-work-with 0.01.*2218

*OK, so it rounds approximately to 0.01; so let's see what is the average rate of change.*2222

*The average rate of change, then, between 0 and 0.01 seconds, is going to be d(0.01) - d(0) over 0.01 - 0.*2228

*That equals...oops, sorry, my mistake: 0.01 is not actually what it came out to be when we put it in the calculator.*2245

*I mis-rounded that just now; it was 0.09951, so if it is 0.09951, if we are going to round that*2253

*to the much-more reasonable-to-work-with thing, we actually get approximately 0.1.*2265

*So, it is not 0.01; 0.01 is still on the bottom, but the top is going to end up coming out to be 0.1 - 0, divided by 0.01; sorry about that.*2271

*It is important to be careful with your rounding.*2288

*That comes out to be 0.1 over 0.01, which comes out to be 10 meters per second.*2290

*And now, you probably haven't taken physics by this point; but if you had, you would actually know that -4.9t ^{2}...*2298

*that is the thing that says the amount that gravity affects where its location is.*2307

*The 10t is the amount of the starting velocity of the ball.*2312

*The ball gets thrown up at 10 meters per second, so it makes sense that its average speed*2315

*between 0 and 0.01--hardly any time to have changed its speed--is going to be pretty much what its speed started at.*2320

*That 10 meters per second is actually showing up there.*2327

*So, there is a connection here between understanding what the physics going on is and the math that is connecting to it.*2329

*All right, finally, between 0 and 2.041 seconds...let's plug in d(2.041) = -4.9(2.041) ^{2} + 10(2.041).*2335

*So, that is going to come out to be -0.0018; so it seems pretty reasonable to just round that to a simple 0.*2354

*Now, what does that mean? That means, at the moment, 2.041 seconds--that is when the ball hits the ground.*2365

*It goes up at 0, and then it comes back down.*2371

*And at 2.041 seconds after having been thrown up, it hits the ground precisely at 2.041 seconds.*2374

*So, 2.041 seconds--then it has a 0 height; so what is its average velocity between 0 and 2.041 seconds?*2380

*Location at 2.041 minus location at 0, divided by 2.041 - 0, equals 0 minus 0, over 2.041, which equals 0 meters per second,*2388

*which makes sense: if we throw the ball up, and then we look at the time when it hits the ground again,*2404

*well, on average, since it went up and it went down, it had no velocity,*2410

*because the amount of time that it has positive velocity going up and the amount of time that it has negative velocity going down--*2415

*it has cancelled itself out, because on average, between the time of its starting on the ground*2420

*and ending on the ground, it didn't go anywhere.*2425

*So, on average, its velocity is 0, because it didn't make any change in its location; great.*2427

*The next example--Example 3: Find the zeroes of f(x) = 3 - |x + 3|.*2433

*Remember: zeroes just mean when f(x) = 0; so we can just plug in 0 = 3 - |x + 3|.*2439

*So, we have |x + 3| = 3; we just add the absolute value of x + 3 to both sides.*2451

*We have |x + 3| = 3; that is what we want to know to figure out when the zeroes are.*2458

*When is this true? Remember, absolute value of -2 is equal to 2, which is also equal to the absolute value of positive 2.*2462

*So, the absolute value of x + 3...we know that, inside of it, since there is a 3 over here...*2474

*there could be a 3, or there could be a -3.*2480

*So, inside of that absolute value, because we know it is equal to 3, we know that there has to currently be a 3, or there has to be a -3.*2484

*We aren't sure which one, though; so we split it into two different worlds.*2498

*We split it into the world where there is a positive on the inside, and we split it into the world where there is a negative on the inside.*2502

*In the positive world, we know that what is inside, the x + 3, is equal to a positive 3.*2508

*In the negative world, we know that the x + 3 is equal to a negative 3.*2518

*Now, it could be either one of these; either one of these would be true; either one of these would produce a 0 for the function.*2523

*So, let's solve both of them: we subtract by 3 on both sides over here; we get x = 0.*2529

*We subtract by 3 on both sides over here; we get x = -6.*2535

*So, the two answers for the roots are going to be -6 and 0; that is when the zeroes of f(x) show up.*2538

*The zeroes of f(x) are going to be at x = 0 and x = -6.*2546

*And if we plug either one of those into that function, we will get 0 out of the function.*2549

*The final example: Show that x ^{6} - 4x^{2} + 7 is even;*2556

*show that -x ^{5} + 2x^{3} - x is odd; and show that x + 2 is neither.*2559

*All right, the first thing we want to do is remind ourselves of what it means to be even.*2564

*To be even means that when we plug in the negative version of a number, a -x is the same thing as if we had plugged in the positive x.*2572

*It doesn't have any effect.*2581

*And the odd version...actually, let's put it in a different color, so we can see how all of the problems match up to each other.*2583

*If we do with the odd version, then if we plug in the negative of a number,*2590

*it comes out to be the negative of if we had plugged in the positive version of the number.*2595

*All right, so the first one: Show that x ^{6} - 4x^{2} + 7 is even.*2599

*So, that was really seeing that expression as if it were a function; so let's show this*2605

*by showing that if we plug in -x, it is the same thing as if we plug in positive x.*2609

*On the left, we will plug in -x; -x gets plugged in; it becomes (-x) ^{6} - 4(-x)^{2} + 7 =...*2613

*if we plugged in just plain x, we would have plain x ^{6} - 4x^{2} + 7; great.*2622

*(-x) ^{6}...remember, a negative times a negative cancels out to a positive.*2630

*We have a 6 up here; we are raising it to the sixth power, so we have an even number of negatives.*2634

*Negative and negative cancel; negative and negative cancel; negative and negative cancel.*2642

*That is a total of 6 negatives; they all cancel each other out; so we actually have (-x) ^{6} being the same thing as if we just said x^{6}.*2645

*Minus 4...the same thing here: -x times -x cancels and just becomes plain x ^{2}...plus 7 equals*2652

*x ^{6} - 4x^{2} + 7; it turns out that it has no effect.*2660

*If we plug in a negative x, we get the same thing as if we had plugged in the positive x.*2666

*Plugging in a negative version of a number is the same thing as plugging in the positive version of the number.*2669

*So, it checks out; it is even; great.*2672

*The next one: let's look at odd: -x ^{5} + 2x^{3} - x is odd.*2677

*We will do the same sort of thing: we will plug -x's in on the left side.*2682

*-(-x) ^{5} + 2(-x)^{3} - (-x); what is going to go on the right side?*2686

*Well, remember: if we plug in the negative version of the number, then it is the negative of if we plugged in the positive version of the number.*2696

*So, it is the negative of if we had plugged in the positive version of the number.*2703

*Plugging in the positive version of the number is just if we have the normal x going in: -x ^{5} + 2x^{3} - x.*2706

*All right, so -(-x) ^{5}: well, what happens when we have (-x)^{5}--what happens to that negative?*2714

*Negative and negative cancel; negative and negative cancel; negative--that fifth one, because it is odd, gets left over.*2720

*So, we have negative; and we just pull that negative out--it is the same thing as -x ^{5}.*2727

*Plus 2...once again, it is odd; a negative and a negative cancel; we are left with one more negative, for a total of 3 negatives; we are left with a negative.*2732

*So, we get 2(-x) ^{3} minus...we can pull that negative out, as well...-x...equals...*2739

*let's distribute this negative; so we get...distribute...cancellation...a negative shows up here...cancel;*2747

*we get positive x ^{5} minus 2x^{3} + x.*2754

*So, let's finish up this left side and do cancellations over here as well; positive, positive; this stays negative.*2761

*Positive, positive; so we get x ^{5} - 2x^{3} + x equals the exact same thing over here on the right side.*2767

*It checks out; yes, it is odd; great.*2778

*Finally, let's show that x + 2 is neither; so, to be neither, we have to fail at being this and fail at being this.*2782

*So, to be neither, it needs to fail being odd and being even; it needs to fail even and odd.*2791

*Let's just try plugging in a number; let's try plugging in, say, -2.*2809

*If we look at x = -2, then that would get us -2 + 2, which equals 0.*2814

*Now, what if we plugged in the flip of -2--we plugged in positive 2?*2821

*x = positive 2...we plug that into x + 2, and we will get 2 + 2, which equals 4.*2825

*Now, notice: 0 is not equal to 4; we just failed being even up here,*2832

*because the negative number and the positive version of that number don't produce the same output.*2841

*Plug in -2; you get 0; plug in +2; you get 4; those are totally different things, so we just failed to be even; great.*2846

*Next, we want to show that it is not odd.*2854

*Odd was the property that, if we plug in the negative, it is going to be equal to the negative of the positive one.*2857

*So, 0 is not equal to -4 either, right? If we plug in -2, we get 0, and if we plug in positive 2,*2861

*it turns out that that is not -0, or just 0; it turns out that that is 4.*2869

*So, we fail to be odd as well, because it isn't the case that if we plug in opposite positive/negative numbers,*2873

*we don't get opposite positive/negative results, because 0 is not the opposite of -4; it is just the opposite of 0, so it fails there.*2881

*So, it checks out: that one is neither; great.*2890

*All right, we just learned a whole bunch of different properties; and they will each come up in different places at different times.*2896

*Just remember these: keep them in the back of your mind.*2901

*If you ever need a reminder, come back to this lesson and just refresh what that one meant,*2903

*because they will show up in random places; but they are all really useful.*2907

*And we will see them a lot more as we start getting into calculus.*2910

*Once you actually get to calculus, this stuff, especially the stuff at the beginning of this,*2913

*where we talked about increasing and decreasing and relative maximums and minimums--*2915

*that stuff is going to become so important if you are going to understand why we are talking about it so much right now in this course.*2919

*All right, I hope you understood everything; I hope you enjoyed it; and we will see you at Educator.com later--goodbye!*2924

1 answer

Last reply by: Professor Selhorst-Jones

Sun Aug 11, 2013 11:03 PM

Post by Tami Cummins on August 11, 2013

In the second part of example 2 what about the negative?

1 answer

Last reply by: Professor Selhorst-Jones

Thu Jul 11, 2013 1:36 PM

Post by Montgomery Childs on June 26, 2013

Dear Mr. Jones,

I really appreciate the time you spend on "definitions" of math terms - i have come to realize this is one of the biggest issues i have had over the years - not the math. This helps so much in my understanding of relationships! Very cool!!!