Answer
Inverse: $y=\pm\sqrt{\dfrac{x-5}{4}}$
The inverse is not a function.
Work Step by Step
To find the inverse of the given function, perform the following steps:
(1) Interchange the variables $x$ and $y$:
$x=4y^2+5$
(2) Solve for $y$
\begin{align*}
x-5&=4y^2\\\\
\frac{x-5}{4}&=y^2\\\\
\pm\sqrt{\frac{x-5}{4}}&=y
\end{align*}
Thus, the inverse of the given function is $y=\pm\sqrt{\dfrac{x-5}{4}}$.
Since for every value of $x$, there corresponds two different values of $y$, then the inverse is NOT a function.