Question

Given f of x find g of z and h of x such that

Answer 1

ANSWER

$\begin{equation*} \begin{cases}{g(x)=\sqrt[3]{x}}+1 \\ {} \\ {h(x)=4x^2-2}\end{cases} \end{equation*}$

EXPLANATION

If f(x) is,

$f(x)=\sqrt[3]{4x^2-2}+1$

There are many possibilities for functions g(x) and h(x) such that f(x) = g(h(x)). One of them is that g(x) is,

$g(x)=\sqrt[3]{x}+1$

And h(x) is,

$h(x)=4x^2-2$

This way, in the composition g(h(x)), when we replace x in g(x) with h(x) we will get function f(x),

$g(h(x))=\sqrt[3]{h(x)}+1=\sqrt[3]{4x^2-2}+1$

Hence, one of the possible equations for g(x) and h(x) such that f(x) = g(h(x)) is:

$\begin{cases}{g(x)=\sqrt[3]{x}}+1 \\ {} \\ {h(x)=4x^2-2}\end{cases}$