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Even/odd composite functions

  1. Apr 8, 2008 #1
    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.
     
    Last edited: Apr 8, 2008
  2. jcsd
  3. Apr 8, 2008 #2
    Consider the fact that the identity function I(x) = x is odd and the absolute value function A(x)=|x| is even.
     
  4. Apr 8, 2008 #3

    HallsofIvy

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    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]?
     
  5. Apr 10, 2008 #4
    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,
     
    Last edited: Apr 10, 2008
  6. Apr 10, 2008 #5

    tiny-tim

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    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:
     
  7. Apr 10, 2008 #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?
     
  8. Apr 10, 2008 #7

    tiny-tim

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    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)?
     
  9. Apr 11, 2008 #8

    HallsofIvy

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

    So you have just said, (f.g)(-x)= f.g(x), haven't you?

     
  10. Apr 11, 2008 #9
    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:
     
  11. Apr 11, 2008 #10

    HallsofIvy

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    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".
     
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