Prove $(g+h)\circ f=g\circ f+ h\circ f$

Question

Let $g,h,f$ be functions with domains and ranges on the real numbers.

I have to prove that $$(g+h)\circ f=g\circ f + h\circ f$$

It seems so simple, but I don't know where to start the proof. Maybe just show that for two step chains of multiplication, addition and exponentiation that this holds? I don't know how to do it in a very elegant way.

Answer 1

Hint:

• The definition of $(g+h)$ is that $(g+h)(x) = g(x)+h(x)$ for every real $x$.
• The definition of $f\circ g$ is that $(f\circ g)(x)=f(g(x))$ for every real $x$.

Also:

To prove that a function $F$ is equal to a function $G$, you need to show that if you take an arbitrary $x\in\mathbb R$, you have $F(x)=G(x)$. All you need to do is use the definitions.