Manipulating Radical Functions

We want to determine whether the given pair of functions are inverse functions. $$\begin{array}{c}f(x)=2\sqrt{x-5}\text{and}g(x)=\frac{1}{4}{x}^2-5\end{array}$$ To do so, we need to verify that the compositions of $f(x)$ and $g(x)$ are the identity function. Since $g(x)$ is a radical function, we first need to find the domain, which only includes values for which the radicand is non-negative. $$\begin{array}{c}x-5\ge 0\iff x\ge 5\end{array}$$ Since the domain of $g(x)$ is all real numbers greater than or equal to $5$, we will only consider them for the $x\text{-}$variable.

Calculating $f(g(x))$

To find the expression for $f(g(x))$ we will start by substituting $\frac{1}{4}{x}^2-5$ for $g(x).$ $$f(g(x))\Rightarrow f\left(\frac{1}{4}{x}^2-5\right)$$ Now we apply the definition of $f(x).$ $$\begin{array}{c}f(x)=2\sqrt{x-5}\\ \Downarrow \\ f\left(\frac{1}{4}{x}^2-5\right)=2\sqrt{\frac{1}{4}{x}^2-5-5}\end{array}$$ Finally, let's simplify and see if the function is the identity function. Since $f(g(x))$ is not the identity function, $f(x)$ and $g(x)$ are not inverse functions.