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Lecture Comments (4)

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!!!

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 x1 and x2 with the formula
    f(x2) − f(x1)

    x2 −x1
  • 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
    f(−x) = f(x).
    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).
  • An odd function (totally different from being an odd number) is one where
    f(−x) = − f(x).
    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-right 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 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

Transcription: Properties of Functions

Hi--welcome back to

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 understand0332

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 flipping0453

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) = x2 - 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, -x4 + 2x3 + 5x2 - 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 my, with yellow, it has 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 points0906

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 discussed0976

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, x1 and x2,1061

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 heights1088

y2 and y1, 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 x2 - x1; and our vertical distance is y2 - y1.1104

So, y1 is the height over here; y2 is the height over here.1115

y2 - y1 over x2 - x1 is the rise, divided by the run.1119

But what are y1 and y2?--if we want to figure out what y1 and y2 are,1127

well, we just need to look at what x1 and x2 are.1132

So, since x1 and x2 are coming to get placed by the function,1134

then y2's height is really just f(x2), 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 y1 over here is from f(x1).1151

So, since our original slope formula is y2 - y1 over x2 - x1,1156

and we know that y1 is the same thing as f(x1) and y2 is the same thing as f(x2),1160

we can just plug those in, and we get the change in our function outputs, f(x2) - f(x1),1164

divided by our horizontal distance, x2 - x1.1171

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 see1312

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 knowing1319

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) = x2: 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 32, 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 exponents1457

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) = x3 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 outputs1549

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 -1.5x4 + x3 + 4x2 + 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 trust1807

that it is going to keep going down, because we have -1.5x4); it is increasing up 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 up1824

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 -4.9t2 + 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 t1 to time t2, it is going to be the location at time t2,2084

minus the location at time t1, over the difference in the time, t2 - t1.2095

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, t2 - t1...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 12, 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 up2211

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 that2253

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 -4.9t2...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 speed2315

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 ground2420

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 x6 - 4x2 + 7 is even;2556

show that -x5 + 2x3 - 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 x6 - 4x2 + 7 is even.2599

So, that was really seeing that expression as if it were a function; so let's show this2605

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 x6 - 4x2 + 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 x6.2645

Minus 4...the same thing here: -x times -x cancels and just becomes plain 7 equals2652

x6 - 4x2 + 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: -x5 + 2x3 - 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: -x5 + 2x3 - 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 -x5.2727

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 x5 minus 2x3 + 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 x5 - 2x3 + 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 later--goodbye!2924