To solve the problem step by step:

Part a: Find the inverse function

The original function is given as:
$f(x) = x + 8$
To find the inverse $f^{-1}(x)$:

1. Replace $f(x)$ with $y$:
   $y = x + 8$
2. Swap $x$ and $y$ to solve for $y$:
   $x = y + 8$
3. Solve for $y$:
   $y = x - 8$ Thus, the inverse function is:
   $f^{-1}(x) = x - 8$

This matches choice C.

Part b: Verify the inverse function

To verify $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$:

1. First verification $f(f^{-1}(x))$:
   Substitute $f^{-1}(x) = x - 8$ into $f(x)$:
   $f(f^{-1}(x)) = f(x - 8) = (x - 8) + 8$ Simplify:
   $f(f^{-1}(x)) = x$

2. Second verification $f^{-1}(f(x))$:
   Substitute $f(x) = x + 8$ into $f^{-1}(x)$:
   $f^{-1}(f(x)) = f^{-1}(x + 8) = (x + 8) - 8$ Simplify:
   $f^{-1}(f(x)) = x$

Both verifications are correct.

Final Answers:

• Inverse Function: $f^{-1}(x) = x - 8$ (choice C).
• Verification: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Do you need more clarification or further steps? Let me know!
Here are some related questions for practice:

1. How do you verify an inverse function for a quadratic equation?
2. What are the steps to find the inverse of $f(x) = 2x + 3$?
3. Can all functions have an inverse? Explain why or why not.
4. What happens to the graph of a function when you find its inverse?
5. How does the horizontal line test relate to the existence of an inverse function?

Tip: To find an inverse, swap $x$ and $y$, then solve for $y$.