Is it true that $(f\cdot g)\circ \phi=(f\circ \phi)\cdot(g\circ \phi)$?

Question

Let be $f,g,\phi: \mathbb{R} \longrightarrow \mathbb{R}$

($\cdot$ is the product of functions)

Thanks!

score 2 · Answer 1 · answered Mar 16 '15 at 18:23

2

Yes, just use the definition of the composition and the product of functions.

answered Mar 16 '15 at 18:23

Ankara

score 2 · Accepted Answer · answered Mar 16 '15 at 18:29

A common tool to use in this case is to consider what happens for a specific point:

Remember that $(f\cdot g)(x) := f(x)\cdot g(x)$

and that $(f\circ \phi)(x) := f(\phi(x))$

Combine these pieces of information to get: $((f\cdot g)\circ\phi)(x) = (f\cdot g)(\phi(x)) = f(\phi(x))\cdot g(\phi(x))=\cdots$

If for every point in the domain it turns out that the two functions applied to that point are equal, then the two functions are equal.

2 Answers2