Let f be any real function and let g be a function given by

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, $g o f=f+f$

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