Please solve the equality of this function. Let $f,g,h:\mathbb R\to \mathbb R$. Show that:
$$
(f+g)\circ h = f\circ h + g \circ h
$$  
$$
(f\cdot g)\circ h = (f\circ h)\cdot(g \circ h)
$$
I know that $(f+g)(x)=f(x)+g(x)$.
But I don't know how to start. 
 A: It is enough to expand the definitions, for example:
\begin{align}
(f\cdot g)(x) &= f(x) \cdot g(x) \tag{$\spadesuit$} \\
(f \circ g)(x) &= f\big(g(x)\big) \tag{$\clubsuit$} \\
\big((f\cdot g)\circ h\big)(x) 
  &\stackrel{\clubsuit^\to}= (f \cdot g)\big(h(x)\big) \\
  &\stackrel{\spadesuit^\to}= f\big(h(x)\big)\cdot g\big(h(x)\big) \\
  &\stackrel{\clubsuit^\gets}= (f \circ h)(x) \cdot (g \circ h)(x) \\
  &\stackrel{\spadesuit^\gets}= \big((f\circ h) \cdot (g \circ h)\big)(x)
\end{align}
For the first equation it is almost the same.
I hope this helps $\ddot\smile$
A: Two functions $f$ and $g$ are defined to be equal when $f(x) = g(x)$ for every $x$ in their domain.
In your exercise, to show that $(f + g) \circ h = f \circ h + g \circ h$, you need to show that
$$((f+g) \circ h)(x) = (f \circ h + g \circ h)(x)$$
for all $x \in \mathbb{R}$.
If you systematically use the general definitions of $f+g$, $fg$ and $f \circ g$, then this should be straightforward. For instance, using the definitions we get that
$$((f+g) \circ h)(x) = (f+g)(h(x)) = f(h(x)) + g(h(x))$$
