The function provided in the question is:

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

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

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

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

Step 2: Swap $x$ and $y$ (this represents finding the inverse)

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

Step 3: Solve for $y$

First, add 2 to both sides: $x + 2 = \frac{y}{5}$

Now, multiply both sides by 5: $5(x + 2) = y$

Thus, the inverse function is: $f^{-1}(x) = 5(x + 2)$

Conclusion:

The correct answer is Option A: $f^{-1}(x) = 5(x + 2)$.

Would you like more details, or do you have any questions?

Here are five related questions:

1. How do you check if a function and its inverse are correct?
2. What are the key properties of inverse functions?
3. How do you find the inverse of more complex functions, like quadratics?
4. How do inverse functions relate to one-to-one functions?
5. What is the graphical interpretation of a function and its inverse?

Tip: Always remember to swap $x$ and $y$ when finding the inverse of a function!