To find the inverse of the function $f(x) = \frac{1}{4}x + 3$, we will follow these steps:

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

$y = \frac{1}{4}x + 3$

2. Swap $x$ and $y$:

$x = \frac{1}{4}y + 3$

3. Solve for $y$:

First, subtract 3 from both sides:

$x - 3 = \frac{1}{4}y$

Next, multiply both sides by 4 to isolate $y$:

$4(x - 3) = y$

Simplify the right-hand side:

$y = 4(x - 3)$

Thus, the inverse function is:

$f^{-1}(x) = 4(x - 3)$

Final Answer:

$f^{-1}(x) = 4x - 12$

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Would you like a detailed breakdown of any steps or additional examples? Here are some related questions:

1. How can you verify if two functions are truly inverses of each other?
2. What happens if the original function involves more complex operations, like quadratics or trigonometric functions?
3. How do you graph both a function and its inverse on the same set of axes?
4. How do you handle finding the inverse when a function is not one-to-one?
5. What are some other methods for solving inverse functions besides algebraic manipulation?

Tip: Always check your work by composing the function with its inverse. If you get $x$ back, you've done it correctly.