Is the Composition of Even and Odd Functions Always Even?

In summary: I chose "doesn't it" because I thought it might be a little clearer to a non-native speaker. You are probably right that I should have chosen "isn't it" to be more consistent with "f(g(x)) is equal to f.g(x)".In summary, we discussed the composition of two functions ƒ · g and how to determine if it is even. We found that if g is even, then ƒ · g is also even, and if g is odd and ƒ is even, then ƒ · g is also even. We also discussed using definitions to prove these statements and considered the example of g(3) = 7 to understand the concept better. Additionally,
  • #1
roam
1,271
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Considering the composition of two functions ƒ · g

If g is even then does this mean that ƒ · g is even? why?

Or if g is odd and ƒ is even, then ƒ · g is even?

How can we show these statements?

Thanks.
 
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  • #2
Consider the fact that the identity function I(x) = x is odd and the absolute value function A(x)=|x| is even.
 
  • #3
Just like you would do any proof of this sort: use the definitions.

If g is even, then g(-x)= g(x). Now, what can you say of [itex]f\cdot g(-x)= f(g(-x))[/itex]?

If g is odd then g(-x)= -g(x). If f is even then f(-x)= f(x). Now, what can you say of [itex]f\cdot g(-x)= f(g(-x))[/itex]?
 
  • #4
If g is even, then g(-x)= g(x). Now, what can you say of [itex]f\cdot g(-x)= f(g(-x))[/itex]?

Could you explain a little bit more on this part please? Thanks.

f(-x)= f(x)
g(-x)= g(x)

f.g(-x) = f(-x(g(-x)))
f.g(-x) = f(g(x))

It is even? Because a function is even if:
f:(-a,a) -> R if for all [tex]x \in (-a,a)[/tex], f(x) = f(-x)

Please help me, I don't know if I'm right.

Regards,
 
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  • #5
Hi roam! :smile:

You need (for each part) a proof that starts "(ƒ · g)(-x) = … ", and finishes " … = (ƒ · g)(x)."

Hint: suppose g(3) = 7.

If g is even, what are (ƒ · g)(3) and (ƒ · g)(-3)?

If g is odd, what are (ƒ · g)(3) and (ƒ · g)(-3)? :smile:
 
  • #6
If g is even, what are (ƒ · g)(3) and (ƒ · g)(-3)? even

If g is odd, what are (ƒ · g)(3) and (ƒ · g)(-3)? even

What if g is odd and f is even? would the (f · g) be even?
 
  • #7
Hi roam! :smile:

No … you're missing the point!

Follow the hint … if g(3) = 7, what is (ƒ · g)(3) (how is (ƒ · g)(3) defined? and so what is it)? And what is (ƒ · g)(-3) (same procedure)?
 
  • #8
roam said:
Could you explain a little bit more on this part please? Thanks.

f(-x)= f(x)
g(-x)= g(x)

f.g(-x) = f(-x(g(-x)))
NO! f.g(-x)= f(g(-x)) as you say below:
f.g(-x) = f(g(x))
and f(g(x))= f.g (x) doesn't it?

It is even? Because a function is even if:
f:(-a,a) -> R if for all [tex]x \in (-a,a)[/tex], f(x) = f(-x)
So you have just said, (f.g)(-x)= f.g(x), haven't you?

Please help me, I don't know if I'm right.

Regards,
 
  • #9
HallsofIvy said:
and f(g(x))= f.g (x) doesn't it?

Sorry for the off-topic:

Is the "=" sign typically read as "is equal to" or as "equals"...?
I thought it would be the former but according to your question tag "doesn't it" you seem to use the latter.
Hm, probably both are possible:smile:
 
  • #10
Actually, I debated whether to say "doesn't it" or "isn't it" myself! Yes, "=" can be read as either "equals" or "is equal to".

I was thinking "f(g(x) equals f.g(x) doesn't it" but I considered "f(g(x)) is equal to f.g(x) isn't it".
 

What are even/odd composite functions?

Even/odd composite functions are functions that are formed by composing an even function with an odd function or vice versa. This means that the input for the composite function is first passed through one function and then through the other function.

What are some examples of even/odd composite functions?

Examples of even/odd composite functions include sin(x)cos(x), x^2sin(x), and tan(x)cos(x).

What is the difference between even and odd functions?

Even functions have the property that f(-x) = f(x), meaning that the function is symmetric about the y-axis. Odd functions have the property that f(-x) = -f(x), meaning that the function is symmetric about the origin.

How do you determine if a composite function is even or odd?

To determine if a composite function is even or odd, you can use the properties of even and odd functions. If the outer function is even and the inner function is odd, the composite function will be even. If the outer function is odd and the inner function is even, the composite function will be odd.

Why are even/odd composite functions important in mathematics?

Even/odd composite functions are important because they allow us to simplify complex functions and make them easier to work with. They also have many applications in physics and engineering, particularly in signal processing and Fourier analysis.

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