Question

Decide whether or not the functions are inverses of each other.f(x) = x^3 - 3, g(x) = ^3sqrt x+3

Answer 1

Given the functions:

`\begin{gathered} f\mleft(x\mright)=x^3-3 \\ g\mleft(x\mright)=^3\sqrt[]{x+3} \end{gathered}`

The functions are inverses of each other when:

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

so, we will find (fog)(x) and compare the result with the previous condition

So, We will substitute the function g(x) into the function f(x) instead of (x)

`\begin{gathered} (f\circ g)(x)=(^3\sqrt[]{x+3})^3-3 \\ \end{gathered}`

simplify the function:

`\begin{gathered} (f\circ g)(x)=(x+3)-3 \\ (f\circ g)(x)=x+3-3 \\ \\ (f\circ g)(x)=x \end{gathered}`

So, the final result is the same as the condition

so, the answer will be:

The functions are inverses of each other.