Please solve the equality of this function.

Question

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.

Answer 1

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 $\ddot\smile$

Answer 2

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