The problem asks us to find the inverse function $f^{-1}(y)$ given the function $f(x) = 2(x+2) + 5$.

1. Problem Description

The problem asks us to find the inverse function $f^{-1}(y)$ given the function $f(x) = 2(x+2) + 5$.

2. Solution Steps

First, we simplify the given function $f(x)$:

$f(x) = 2(x+2) + 5 = 2x + 4 + 5 = 2x + 9$.

To find the inverse function, we replace $f(x)$ with $y$ and then swap $x$ and $y$:

$y = 2x + 9$.

Swapping $x$ and $y$, we get:

$x = 2y + 9$.

Now, we solve for $y$:

$x - 9 = 2y$

$y = \frac{x-9}{2}$.

Finally, we replace $y$ with $f^{-1}(x)$. Since we want the inverse in terms of $y$, we have $f^{-1}(y) = \frac{y-9}{2}$.

3. Final Answer

$f^{-1}(y) = \frac{y-9}{2}$