To find the inverse of the function $f(x) = 5x + 7$, follow these steps:

1. Replace $f(x)$ with $y$: $y = 5x + 7$

2. Swap $x$ and $y$:
   The inverse function is found by interchanging $x$ and $y$. $x = 5y + 7$

3. Solve for $y$:
   Isolate $y$ to express it in terms of $x$. $x - 7 = 5y$ $y = \frac{x - 7}{5}$

4. Replace $y$ with $f^{-1}(x)$: The inverse function is: $f^{-1}(x) = \frac{x - 7}{5}$

Thus, the inverse of the function $f(x) = 5x + 7$ is: $f^{-1}(x) = \frac{x - 7}{5}$

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. What is the graphical interpretation of a function and its inverse?
2. How do you verify if two functions are inverses of each other?
3. How does the domain and range of a function change when finding its inverse?
4. What are some common mistakes to avoid when solving for the inverse of a function?
5. Can all linear functions have inverses?

Tip: To check if you found the correct inverse, apply both functions and ensure $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.