Question

Please help me find the inverse

Answer 1

$f^{-1}(x)$ is supposed to be a function such that

$f^{-1}(f(x))=x$

In this case, we need

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

To recover $x$ from $\sqrt[3]{x-2}$, we would first need to raise $\sqrt[3]{x-2}$ to the third power:

$(\sqrt[3]{x-2})^3=x-2$

Then add 2:

$(x-2)+2=x$

To recap, we carried out

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

which implies that the inverse function is

$f^{-1}(x)=x^3+2$

To verify: we should also have that $f(f^{-1}(x))=x$. We get

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