Question

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

Answer 1

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

Step-by-step 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}`