Answer
Inverse: $y=\pm\sqrt{5-x}$
The inverse is NOT a function.
Work Step by Step
To find the inverse of the given function, perform the followiong steps:
(1) Interchange $ x$ and $y$.
$$x=-y^2+5$$
(2) Solve for $y$.
\begin{align*}
x-5&=-y^2 &\text{Subtract 5 from each side.}\\\\
-1(x-5)&=-1(-y^2) &\text{Multiply $-1$ to both sides.}\\\\
-x+5&=y^2\\\\
5-x&=y^2\\\\
\pm\sqrt{5-x}&=\sqrt{y^2} &\text{Take the square root of both sides.}\\\\
\pm\sqrt{5-x}&=y
\end{align*}
Note that for $y=\pm\sqrt{5-x}$, for every value of $x$, there are corresponds two different values of $y$.
Thus, the inverse of the given function is NOT a function.
