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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!

Which function represents the inverse of function f(x) = 3x + 5?

Explore how to find the inverse of the linear function f(x) = 3x + 5 with a detailed, step-by-step solution. Understand the process of swapping x and y to solve for the inverse, suitable for high school algebra students.

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