To find the inverse of the function $y = 4x - 7$, follow these steps:

1. Replace $y$ with $x$ to set up the equation for the inverse: $x = 4y - 7$

2. Solve for $y$: $x + 7 = 4y$ $y = \frac{x + 7}{4}$

3. Write the inverse function in the desired form $ax + b$: $f^{-1}(x) = \frac{1}{4}x + \frac{7}{4}$

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

Would you like additional explanations or have any questions? Here are some related questions for further practice:

1. How would you verify if two functions are inverses of each other?
2. What is the graph of the original function $y = 4x - 7$ and its inverse?
3. Can a function have an inverse if it is not one-to-one? Why or why not?
4. How does the slope of a function compare to its inverse in linear equations?
5. What happens to the domain and range of a function when finding its inverse?

Tip: To find an inverse function, always swap $x$ and $y$, then solve for $y$. This ensures you're correctly reversing the operation.