## Constant functions

Forum for the GRE subject test in mathematics.
Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

### Constant functions

If f(g(x)) is a constant, is it necessary that either f or g is a constant function?

yoyobarn
Posts: 80
Joined: Sun Dec 19, 2010 7:01 am

### Re: Constant functions

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.

Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

### Re: Constant functions

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

owlpride
Posts: 204
Joined: Fri Jan 29, 2010 2:01 am

### Re: Constant functions

Or g = f^{-1}.

blitzer6266
Posts: 61
Joined: Sun Apr 04, 2010 1:08 pm

### Re: Constant functions

The identity? (gotta be honest, that was the first thing that came to my mind too.. not right though)

owlpride
Posts: 204
Joined: Fri Jan 29, 2010 2:01 am

### Re: Constant functions

Ooops. Lol!

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