The question asks for the inverse of the function $f(x) = 4x - 6$. Let's find the inverse function $f^{-1}(x)$.

Steps to find the inverse:

1. Start with the function: $y = 4x - 6$

2. Swap $x$ and $y$: $x = 4y - 6$

3. Solve for $y$: $x + 6 = 4y$ $y = \frac{x + 6}{4}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x + 6}{4}$

Now, let's compare this with the answer choices:

• a. $\frac{1}{4}(6 - x)$
• b. $\frac{1}{4}(6 + x)$
• c. $-\frac{1}{4}(x - 6)$
• d. $\frac{1}{4}(x + 6)$
• e. $-\frac{1}{4}(6 - x)$

The correct answer is d. $\frac{1}{4}(x + 6)$.

Would you like any further clarification or details on this?

Here are five related questions to practice:

1. What is the inverse of the function $f(x) = 3x + 5$?
2. How do you verify that two functions are inverses of each other?
3. If $f(x) = 2x - 4$, what is $f^{-1}(x)$?
4. What happens to the graph of a function when you take its inverse?
5. Can all functions have an inverse? Why or why not?

Tip:

When finding the inverse of a function, always remember to swap $x$ and $y$, and then solve for $y$.