I would like some help with proving or disproving the next statements:
- If $f(x)$ and $g(x)$ are defined on the same domain $D$ than $f\circ g$ and $g\circ f$ are also defined on D.
I think this is not true but only if I can look on part of the domain. for example if $D$ is {$x\in \mathbb R| x\ne0$} and $f(x)=\frac{1}{x}$ $g(x)=\sin x$ than $f\circ g$ is not defined for every $n\pi$ for every natural n. but can it be false if you look at two functions with exactly the same domain?
- If $f(x)$ and $g(x)$ are defined for every $\mathbb R$ than $f\circ g$ and $g\circ f$ are also defined for every $\mathbb R$
I think this is true since for every $x$ I will input $f(x)$ is defined and for every $x$ I will input in $g(x)$ it will also be defined so also $f\circ g$ will be defined.
- If $f\circ g =g\circ f$ than $f(x) = g(x)$
I think this is false. for example: $f(x) = -x$ and $g(x)=x^3$
- If $f\circ f=g\circ g$ than $f(x)=g(x)$
I think this is false. for example $f(x)=\frac{1}{x}$ and $g(x)=\frac{2}{x}$
My main problem is I'm not sure how to approach proving the true statements.
Any help will be highly appreciated.
If $f,g:\Bbb R\to\Bbb R$, then since $f(x),g(x)\in\Bbb R$ for all $x\in\Bbb R$, we have that $f(g(x))$ and $g(f(x))$ are defined for all $x\in\Bbb R$.
This is, in general, not true if $f$ and $g$ do not have the same codomain.