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Constant functions

Posted: Wed Nov 09, 2011 10:11 am
by Hom
If f(g(x)) is a constant, is it necessary that either f or g is a constant function?

Re: Constant functions

Posted: Wed Nov 09, 2011 10:39 am
by yoyobarn
I think it is not necessarily so.

Example: $$f(x)=e^{ix}$$
$$g(x)=2x\pi$$, if x is an integer, and 0 otherwise.

Then, $$f(g(x))=1$$ but neither f nor g are constant functions.

Re: Constant functions

Posted: Wed Nov 09, 2011 10:53 am
by Hom
yoyobarn wrote:I think it is not necessarily so.

Example: $$f(x)=e^{ix}$$
$$g(x)=2x\pi$$, if x is an integer, and 0 otherwise.

Then, $$f(g(x))=1$$ but neither f nor g are constant functions.
Thank you yoyobarn. You're right. You inspired me to think of another case in R.
f(x) = sinx
g(x)= [x]*pi
f(g(x))=0

Re: Constant functions

Posted: Thu Nov 10, 2011 2:51 am
by owlpride
Or g = f^{-1}.

Re: Constant functions

Posted: Thu Nov 10, 2011 3:57 am
by blitzer6266
The identity? (gotta be honest, that was the first thing that came to my mind too.. not right though)

Re: Constant functions

Posted: Thu Nov 10, 2011 4:26 am
by owlpride
Ooops. Lol!

That's what happens when you use 1_X to denote the identity function on X...