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

Given: $f(x) = 4x - 7$

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

$y = 4x - 7$

Step 2: Swap $x$ and $y$ to find the inverse

$x = 4y - 7$

Step 3: Solve for $y$

$x + 7 = 4y$ $y = \frac{x + 7}{4}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x + 7}{4}$

Final Answer:

$f^{-1}(x) = \frac{x + 7}{4}$

Would you like a more detailed explanation or have any questions?

Here are some related questions to expand your understanding:

1. How can we verify that $f(x)$ and $f^{-1}(x)$ are true inverses of each other?
2. What would happen if $f(x) = 2x + 3$? How do we find its inverse?
3. How can we graph $f(x)$ and its inverse on the same coordinate plane?
4. What is the domain and range of $f(x)$ and $f^{-1}(x)$?
5. What are the geometric properties of inverse functions?

Tip: Always check that the composition $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ to confirm that the functions are true inverses.