Example: Find the inverse function of \(f(x)=\sqrt[3]{x^5-7}+8\). Then verify the correctness of your work.
Solution:
$$
\begin{align}
y &= \sqrt[3]{x^5-7}+8, \,\,\, \text{then swap the variables}\\
x &= \sqrt[3]{y^5-7}+8, \,\,\, \text{next solve for } y\\
x-8 &= \sqrt[3]{y^5-7}\\
(x-8)^3 &= y^5-7 \\
(x-8)^3+7 &= y^5 \\
\sqrt[5]{(x-8)^3+7} &= y, \,\,\, \text{finally, declare victory by saying that}\\
f^{-1}(x) &= \sqrt[5]{(x-8)^3+7} \,\,\, \checkmark
\end{align}
$$
Verification of correctness:
$$
\begin{align}
(f^{-1}\circ f)(x) &= f^{-1}(f(x)) \\
&= f^{-1}\left(\sqrt[3]{x^5-7}+8\right) \\
&= \sqrt[5]{\left(\sqrt[3]{x^5-7}+8-8\right)^3+7} \\
&= \sqrt[5]{\left(\sqrt[3]{x^5-7}\right)^3+7} \\
&= \sqrt[5]{x^5-7+7} \\
&= \sqrt[5]{x^5} \\
&= x
\end{align}
$$
Also, the opposite order
$$
\begin{align}
(f\circ f^{-1})(x) &= f(f^{-1}(x)) \\
&= f\left( \sqrt[5]{(x-8)^3+7} \right) \\
&= \sqrt[3]{\left( \sqrt[5]{(x-8)^3+7} \right)^5-7}+8\ \\
&= \sqrt[3]{\left( (x-8)^3+7 \right)-7}+8\ \\
&= \sqrt[3]{ (x-8)^3 }+8\ \\
&= x-8+8\\
&= x
\end{align}
$$