Question:
Let $f$ be any real function and let $g$ be a function given by $g(x)=2 x$. Prove that $g$ of $=f+f$.
Solution:
Given, $f: R \rightarrow R$
Since $\mathrm{g}(x)=2 x$ is a polynomial, $g: R \rightarrow R$
Clearly, gof $: R \rightarrow R$ and $f+f: R \rightarrow R$
So, domains of gof and $\mathrm{f}+\mathrm{f}$ are the same.
$($ gof $)(x)=g(f(x))=2 f(x)$
$(f+f)(x)=f(x)+f(x)=2 f(x)$
$\Rightarrow(g o f)(x)=(f+f)(x), \forall x \in R$
Hence, $g o f=f+f$