Analyzing functions
Connection between even and odd numbers and functions A possible reason why even functions are called "even" and odd functions are called "odd"
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- In the last video on even and odd functions,
- I talked about how
- you shouldn't get confused between
- even functions and even numbers
- and odd functions and odd numbers.
- And I said that there wasn't any obvious connection
- between the words 'even function'
- and our notion of even numbers,
- or any connection between
- odd functions and odd numbers.
- And, I was wrong!
- There actually is a relatively obvious connection
- and this was pointed out by the youtube user Nothias.
- And the connection,
- I almost explicity did it
- in the last example.
- When I showed an even function,
- I showed you x^2.
- When I showed you an odd function,
- I showed you x^3.
- When I wanted to show you another odd function,
- I showed you y = x, or f(x) = x^1.
- And so you might start to notice
- what Nothias pointed out
- is that these [archetypal], or these good examples,
- or these simple examples
- of even and odd functions
- when I just have a simple 'x' raised to some power.
- Whether the power is even or odd
- is going to tell you
- whether the function is even or odd.
- And you want to be very careful here.
- Not all even or odd functions
- even have exponents in them!
- They could be trigonometric functions;
- they might be some other type of wacky function.
- You don't have to have exponents,
- it's just that these exponents
- are probably where the motivation
- for calling these
- even functions and odd functions came from.
- And let me just be clear:
- it's not just also any polynomial,
- and even in the last video,
- when we had x^3 + 1,
- this was neither
- even [nor] odd.
- But if you just have the pure
- x raised to some power,
- then all of a sudden,
- the motivations for calling [them] even and odd
- start to make sense.
- Because if I have f(x) = x^1,
- that's the same thing as y = x.
- This is odd!
- And it kind of jells with the name
- because we're also raising it
- to an odd power.
- If we have f(x) = x^2,
- we saw in the previous video
- [that] this is even.
- And it kind of jells with the idea
- that we're raising it to an even power.
- I could keep going.
- If it [were] x^3,
- that is odd.
- I could keep going.
- Let me write it this way:
- In general, if you have f(x) = x^n,
- then this is odd if 'n' is odd,
- and this is an even function if 'n' is even.
- And I want to make it very clear here.
- The whole point of this video
- is to clarify the motivation
- for calling them even or odd functions.
- Not all even functions
- are going to be of this form here,
- where it's x raised to some even power,
- and not all odd functions are going to be.
- And I also don't want you to be confused
- that if I have something like
- x^3 then I have other stuff past that,
- then you say," oh, x^3, that's an odd [function]!"
- But this is not an odd function.
- Just when it's just a pure strip-down
- x^3 or x^1,
- can you really make that statement.
- But that really is probably
- where the motivation comes
- for naming them even or odd functions.
- And then the other symmetric functions,
- even if they don't involve an exponent,
- maybe this is some type of trigonometric function,
- you're calling it even because
- it has the same type of symmetry
- as say x^2, or x to an even power.
- So you kind of group them
- all together as even functions.
- And then all of these,
- even though this may or may not
- have an exponent in it,
- it has the same type of symmetries
- as x raised to an odd power.
- So that's why call them odd functions.
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