To find the inverse of the function $f(x) = 4x + 6$, we follow these steps:

Step 1: Set $f(x) = y$

$y = 4x + 6$

Step 2: Solve for $x$ in terms of $y$

• Subtract 6 from both sides: $y - 6 = 4x$
• Divide both sides by 4: $x = \frac{y - 6}{4}$

Step 3: Swap $x$ and $y$ to get the inverse function

$f^{-1}(x) = \frac{x - 6}{4}$

Step 4: Simplify

This can be rewritten as: $f^{-1}(x) = \frac{x}{4} - \frac{6}{4} = \frac{x}{4} - \frac{3}{2}$

Thus, the correct inverse function is: $f^{-1}(x) = \frac{x}{4} - \frac{3}{2}$

This corresponds to the second option in the given choices.

Let me know if you'd like further details or have any questions! Here are some related questions to expand your understanding:

1. What is the geometric meaning of the inverse function?
2. How do you verify that two functions are inverses of each other?
3. What happens to the domain and range when finding the inverse of a function?
4. How do you find the inverse of more complex functions?
5. How can the inverse function be applied to real-world problems?

Tip: Always check if the inverse function makes sense by plugging values from the original function into the inverse, and vice versa.