Manipulating Radical Functions

We want to determine whether the given pair of functions are inverse functions. $$\begin{array}{c}f(x)=\sqrt[5]{\frac{x+9}{5}}\text{and}g(x)=5x^5-9\end{array}$$ To do so, we need to verify that the compositions of $f(x)$ and $g(x)$ are the identity function.

$f(g(x))$

To find the expression for $f(g(x))$ we will start by substituting $5x^5-9$ for $g(x).$ $$f(g(x))\Rightarrow f\left(5x^5-9\right)$$ We will now apply the definition of $f(x).$ $$\begin{array}{c}f(x)=\sqrt[5]{\frac{x+9}{5}}\\ \Downarrow \\ f(5x^5-9)=\sqrt[5]{\frac{5x^5-9+9}{5}}\end{array}$$ Finally, let's simplify and see if the function is the identity function. We found that $f(g(x))$ is the identity function.

$g(f(x))$

We will now investigate the expression $g\left(f(x)\right).$ To find the expression, this time we will start by substituting $\sqrt[5]{\frac{x+9}{5}}$ for $f(x).$ $$g(f(x))\Rightarrow g\left(\sqrt[5]{\frac{x+9}{5}}\right)$$ Now, we will apply the definition of $g(x).$ $$\begin{array}{c}g(x)=5x^5-9\\ \Downarrow \\ g\left(\sqrt[5]{\frac{x+9}{5}}\right)=5{\left(\sqrt[5]{\frac{x+9}{5}}\right)}^5-9\end{array}$$ Finally, let's simplify and see if the function is the identity function. We found that $g(f(x))$ is also the identity function. Therefore, $f(x)$ and $g(x)$ are inverse functions.