To find the inverse of the function $f(x) = 3x + 5$, we need to solve for $x$ in terms of $y$. Here are the steps to find the inverse:

1. Replace $f(x)$ with $y$: $y = 3x + 5$

2. Swap $x$ and $y$: $x = 3y + 5$

3. Solve for $y$: $x - 5 = 3y$ $y = \frac{x - 5}{3}$ Simplifying further: $y = \frac{1}{3}x - \frac{5}{3}$

Therefore, the inverse function is:

$f^{-1}(x) = \frac{1}{3}x - \frac{5}{3}$

Now, let's compare this result with the given options:

• Option A: $s(x) = \frac{1}{3x + 5}$ - This is incorrect.
• Option B: $q(x) = \frac{1}{3}x + \frac{5}{3}$ - This is incorrect.
• Option C: $p(x) = \frac{1}{3}x - \frac{5}{3}$ - This matches our inverse function.
• Option D: $r(x) = -3x - 5$ - This is incorrect.

The correct answer is:

C. $p(x) = \frac{1}{3}x - \frac{5}{3}$

Would you like me to go through any steps in more detail, or do you have any questions? Here are some related questions you might consider:

1. How do you verify if a function and its inverse are correct?
2. What is the graphical interpretation of a function and its inverse?
3. How do you find the inverse of a function involving fractions or more complex expressions?
4. What is the importance of finding the inverse of a function in real-world applications?
5. How do you determine if a function has an inverse without solving?

Tip: Always swap $x$ and $y$ and solve for $y$ to find the inverse of any function!