For the function $f(x) = \frac{1}{4}x - 2$, let's find the inverse step by step.

Steps to Find the Inverse:

1. Rewrite the function with $y$:

   $y = \frac{1}{4}x - 2$

2. Swap $x$ and $y$:

   $x = \frac{1}{4}y - 2$

3. Solve for $y$:

   • Start by isolating the term with $y$:

     $x + 2 = \frac{1}{4}y$

   • Multiply both sides by 4:

     $4(x + 2) = y$

   • Simplify:

     $y = 4x + 8$

The inverse function is:

$f^{-1}(x) = 4x + 8$

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

Related Questions:

1. How can you graph both a function and its inverse?
2. What happens to the domain and range when finding an inverse function?
3. How do you interpret the slope and intercepts of an inverse function?
4. Can the inverse of a linear function be a non-linear function?
5. How does the inverse function relate to reflections over the line $y = x$?

Tip: Always check your inverse by substituting back into the original function to ensure it simplifies to $x$.