To find the inverse of the function $f(x) = 2 - \frac{12}{x+5}$, we follow these steps:

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

$y = 2 - \frac{12}{x+5}$

Step 2: Swap $x$ and $y$.

$x = 2 - \frac{12}{y+5}$

Step 3: Solve for $y$ in terms of $x$.

1. Subtract 2 from both sides: $x - 2 = -\frac{12}{y+5}$

2. Multiply through by $-(y+5)$ to eliminate the fraction: $-(x-2)(y+5) = 12$

3. Distribute the $-(x-2)$: $-(x-2)y - 5(x-2) = 12$

4. Rearrange to isolate $y$: $-(x-2)y = 12 + 5(x-2)$

   Simplify the right-hand side: $-(x-2)y = 12 + 5x - 10$

   $-(x-2)y = 5x + 2$

5. Solve for $y$: $y = \frac{-(5x+2)}{-(x-2)}$

   Simplify the negatives: $y = \frac{5x+2}{x-2}$

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

$f^{-1}(x) = \frac{5x+2}{x-2}$

Final Answer:

The inverse of $f(x) = 2 - \frac{12}{x+5}$ is: $f^{-1}(x) = \frac{5x+2}{x-2}.$

---

Do you want a verification or further explanation?
Here are some related questions to consider:

1. How can we verify that this is the correct inverse?
2. What is the domain and range of $f(x)$ and $f^{-1}(x)$?
3. How do the graphs of $f(x)$ and $f^{-1}(x)$ relate geometrically?
4. What happens to the function at $x = -5$, and how does it affect the inverse?
5. Are there any restrictions on $x$ for the inverse function?

Tip: Always check the domain and range of a function and its inverse to ensure they correspond appropriately.