We are given the function $f(x) = -\frac{1}{2}x + 16$ and we need to find its inverse, $f^{-1}(y)$.

1. Problem Description

We are given the function $f(x) = -\frac{1}{2}x + 16$ and we need to find its inverse, $f^{-1}(y)$.

2. Solution Steps

To find the inverse function $f^{-1}(y)$, we follow these steps:

1. Replace $f(x)$ with $y$:

$y = -\frac{1}{2}x + 16$

2. Swap $x$ and $y$:

$x = -\frac{1}{2}y + 16$

3. Solve for $y$:

$x - 16 = -\frac{1}{2}y$

Multiply both sides by $-2$:

$-2(x - 16) = y$

$y = -2x + 32$

4. Replace $y$ with $f^{-1}(x)$:

$f^{-1}(x) = -2x + 32$

5. Finally, replace $x$ with $y$ to find $f^{-1}(y)$:

$f^{-1}(y) = -2y + 32$

3. Final Answer

$f^{-1}(y) = -2y + 32$