Question

Please help with this I am completely stuck on it

Answer 1

`f(x)=\sqrt[3]{x-4} , g(x)=6x^(2)\textrm{ or }f(x)=\sqrt[3]{x},g(x)=6x^(2) -4`

Explanation:

Given:

The function,
`H(x)=\sqrt[3]{6x^(2)-4}`

Solution 1:

Let
`f(x)=\sqrt[3]{x}`

If
`f(g(x))=H(x)=\sqrt[3]{6x^(2)-4}`, then,

`\sqrt[3]{g(x)} =\sqrt[3]{6x^(2)-4}\\g(x)=6x^(2)-4`

Solution 2:

Let
`f(x)=\sqrt[3]{x-4}`. Then,

`f(g(x))=H(x)=\sqrt[3]{6x^(2)-4}\\\sqrt[3]{g(x)-4}=\sqrt[3]{6x^(2)-4} \\g(x)-4=6x^(2)-4\\g(x)=6x^(2)`

Similarly, there can be many solutions.