Recognizing Odd and Even Functions

Even and odd functions

Let's see if we can learn a thing or two about even, even functions and odd functions.

Even functions and on the right-hand side over here, we'll talk about odd functions.

If we have time we'll talk about functions that are neither even nor odd.

So, before I go into kind of a formal definition of even functions,

I just want to show you what they look like visually,

because I think that's the easiest way to recognize them

and then it'll also make a little more sense when we talk about the formal

definition of an even function.

So, let me draw some coordinate axes here.

X-axis and then, -let me see if I can draw that a little straighter.

Move this right over here, and that is my y-axis

Or I could say y is equal to f(x) axis, just like that.

Let me draw the graph of f(x).

f(x) is equal to x-squared, or Y is equal to x-squared, either one.

So let me draw the first quadrant.

It looks like this.

And then in the second quadrant it looks like this.

It looks like- oh let me try to draw this symmetric.

Pretty good job.

The f(x) is equal to x squared is an even function.

And the way that you recognize it is because it has this symmetry.

around the Y-axis.

If you take- If you take what's going on, on the right-hand side

to the right of the y-axis and you just reflect it over the Y-axis,

you get the other side of the function and that's what tells you

it is an even function.

And I want to show you one interesting property here.

If you take any x-value - let's say you take a positive x-value.

Let's say you take the value x is equal to two.

If you find f(2) you're going to find four.

That's going to be four for this particular function for f(x) where two squared is

four.

And if you took the negative version of two- So if you took negative two

If you took negative two and you evaluated the function there,

you're also- you are also goint to get four, and this,

hopefully, or maybe makes complete sense to you.

You're like, "Well Sal, obviously if I reflect this function over

the Y-axis, that's going to be the case."

Whatever function value I get at the positive value of number,

I'm going to get the same function value at the negative value.

And this is what kind of leads us to the formal definition.

If a function is even, or I could say a function is even

if and only if- So it's even. And don't get confused with the term even function

and the term even number.

They're completely different, um, kind of ideas. So there's- there's not, at

least not an obvious connection, that I know of, between even functions

and even numbers or odd functions and odd numbers.

So you're an even function if and only if, f of- f(x)

is equal to f(-x).

And the reason why I didn't introduce this from the beginning

is because this is really the definiton of even functions.

Because when you look at this you are like:

Hey, what does this mean?

F(x) is equal to f(-x) and all it does mean is this.

It means if I would take f(2)- f(2) is 4.

So let me show you with the particular case.

f(2) is equal to f(-2)- f(-2).

And this particular case for f(x)=f(x^2) they are both equal to 4.

So really, it's just another way of saying that the function can be reflected

or the left side of the function is the reflection of the right side of the function

across the vertical axis, across the y-axis.

And just to make sure we have a decent understanding here

let me draw a few more even functions.

And i'm going to draw some fairly wacky things

just so you would really kinda learn to visually recognize them.

So a function like, let's say like this.

Maybe jumps up to here and does something like that.

And then on this side it does the same thing.

It's the reflection, so it jumps up here.

then it goes like this and then it goes like this.

And i'm trying to draw so they are the mirror image of eachother.

This is an even function.

You take what's going on on the right hand side of this function

and you literally just reflect it over the y-axis and you get the left hand side of the function.

And you could see that even this holds.

If I take some value.

Let's say that this value right here is... I don't know -- 3.

Let's say that the f(3) over here is equal to, let's say that is 5.

So this is 5.

We see that f(-3) is also going to be equal to 5.

And that's what our definition of even function told us.

I can draw, let me just draw one more to really make sure.

I'll do the axis in that same green colour.

Let me do one more like this.

And you could have maybe some type of trigonometric looking function.

That looks like this.

That looks like that.

And it keeps going in either direction.

So something like this would also be even.

So all of these are even functions. Now you are probably thinking.

What is an odd function?

And let me draw an odd function for you.

So let me draw the axis once again.

X-axis, y-axis so the f of x-axis and to show you an odd function.

I'll give you a particular odd function, maybe the most famous of odd functions.

This is probably the most famous of the even functions.

And it is f(x) although there are probably other contendors for the most famous odd function.

f(x) is equal to x^3.

And it looks like and you might have seen the graph of it.

If you haven't you can graph it by trying some points.

It looks like that.

and the way to visually recognize an odd function is you look at what's going on

to the right of the y-axis, once again, this is y-axis, this is the x-axis.

You have all of this business to the right of the y-axis.

If you reflect it over the y-axis you would get something like this.

You would get something like this and if the left side of this graph looked like this

we would be dealing with an even function.

Clearly it doesn't.

To make this an odd function we reflected once over the y-axis

and then reflected the x-axis or another way to think about it

reflected once over the y-axis and then make it negative.

Either way it will get you there. Or you could even reflect it over the x-axis

and then the y-axis, so you are kinda doing two reflections.

So clearly if you take this up here and then you reflect it over the x-axis.

You get these values, you get this part of the graph right over here.

And if you try to do it with a particular point

I'm doing this to kinda hint that with the definition

the formal definition of an odd function this is going to be.

Let's try a point, let's try 2 again.

If you had the point 2, f(2) is 8.

So f(2) is equal to 8.

Now what happens if we take negative 2.

If we take negative 2, f(-2), -2^3 that is just going to be -8.

So f(-2) is equal to -8.

And in general if we take, let me just write it over here.

f(2) so we are just taking one particular example from this particular function.

We have, f(2) is equal to, not f(-2).

8 does not equal -8.

8 is equal to negative of -8.

So that's positive 8.

So f(2) is equal to the negative of f(-2).

We figured out, just I want to make it clear.

We figured out f of 2 is 8. 2^3 is 8.

We know that f of -2 is -8. -2^3 is -8

So you have the negative of -8, negatives cancel out and it works out.

So in general, you have an odd function.

So here is the definition.

You are dealing with an odd function if and only if

f of x for all the x's that are defined on that function or for which that function is defined.

If f(x) is equal to negative of f(-x) or you'll sometimes see it the other way

if you multiply both sides of this equation with -1 you would get

negative of f(x) is equal to f(-x)

and sometimes you will see when it has swapped around

and you will say f of negative x is equal to

Let me write that, careful.

Is equal to -f(x).

I just swapped these two sides.

So let me just draw you some more odd functions.

Some more odd functions.

So I'll do these visually.

So, I'll just draw that a little bit cleaner.

So if you have a...

maybe it looks something... maybe the function does something wacky

Maybe it does something wacky like this on the right hand side,

If it was even you would reflect it there, but we are going to have and odd function

so we are going to reflect it again.

So the rest of the function is going to look like this.

So what i have drawn, the non-dotted lines

this right here is an odd function and you could even look at the definition.

If you'd take some value, a, and then you'd take f(a) which would put you up here.

this right here would be f(a)

if you would take the negative value of that, if you would take -a here

-a, f(-a) is gonna be down here.

So f(-a) is going to be equal to, it's going to be the same distance

from the horizontal axis.

It's not compleatly clear the way I drew it just now.

So it's maybe gonna be like right over here.

So this right over here is going to be f of negative a

which is the same distance from the origin is of f of a, it's just the negative

It's not, I didn't compleatly draw it to scale.

Let me draw one more of these odd functions.

I think you might get the point.

I shall draw a very simple odd function,

just to show you that it doesn't always have to be something crazy.

So a very simple odd function, would be y is equal to x.

y=x, something like this.

Whoops.

Y is equal going through the origin.

You reflect what's on the right onto to the left, you get that

and you reflect it down you get all of this stuff in the third quadrant.

So this is also an odd function.

Now I wanna leave you with a few things that are not odd functions

and at some times might be confused to be odd functions.

So you might have something like this where you have a... maybe have a parabola,

but it doesn't, it's not symmetric around the y-axis and your temptation might be:

Hey, there is this symmetry for this parabola, but it's not

it's not being reflected around the y-axis, you don't have a situation here

where f of x is equal to f of negative x.

So this is not, this is neither.

Neither odd nor even.

Similary you might see, let's say you see a shifted,

a shifted cubic function, so let's say you have something like this.

Let's say you have x to the third plus 1.

so f(x) is equal to x^3+1.

So it might look something like this.

And once again you will be tempted to call this an odd function,

but because it has shifted up it is no longer an odd function.

You could look at that visually.

So this is f of x is equal to x^3+1.

If you take what's on the right hand side and reflect it onto to the left hand side

you would get something like that and then if you reflect that down

you would get something like that.

So this is not an odd function.

You are not... this isn't the left reflection and then the top-bottom reflection

of what's going on on the right hand side.

This over here actually would be.