Analyzing functions
Recognizing Odd and Even Functions Even and odd functions
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- 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.
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