Question

Find two functions F and G such that (f°g) (x) = h(x).

Answer 1

This operation is equivalent to:

`(f\circ g)(x)=f(g(x))`

It is like a function inside a function. So, we can look for parts in h(x) that are common and call that the function inside.

As we can see, h(x) have terms with x + 4, so if we call:

`g(x)=x+4`

We can see that h(x) becomes:

`h(x)=(g(x))^2+2g(x)`

And if we substitute g(x) by x, we will get the expression of the ouside function f(x):

`f(x)=x^2+2x`

This way, we have:

`(f\circ g)(x)=(g(x))^2+2g(x)=(x+4)^2+2(x+4)=h(x)`

So, the functions are:

`\begin{gathered} g(x)=x+4 \\ f(x)=x^2+2x \end{gathered}`