### Problem:
Given the function \( f(x) = 3x + 5 \), find its inverse function \( f^{-1}(x) \).

### Solution:

To find the inverse function \( f^{-1}(x) \), follow these steps:

1. **Start with the function equation:**
   \[
   y = 3x + 5
   \]
   Here, \( y \) represents \( f(x) \).

2. **Swap \( x \) and \( y \):**
   This step is crucial because the inverse function reverses the roles of the input and the output.
   \[
   x = 3y + 5
   \]

3. **Solve for \( y \):**
   Now, we isolate \( y \) to find the inverse:
   \[
   x - 5 = 3y
   \]
   \[
   y = \frac{x - 5}{3}
   \]

4. **Replace \( y \) with \( f^{-1}(x) \):**
   The inverse function is:
   \[
   f^{-1}(x) = \frac{x - 5}{3}
   \]

### Conclusion:
The inverse function of \( f(x) = 3x + 5 \) is \( f^{-1}(x) = \frac{x - 5}{3} \).

### Verification:
To verify the correctness of the inverse, we can check the compositions \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \):

- \( f(f^{-1}(x)) = 3\left(\frac{x - 5}{3}\right) + 5 = x \)
- \( f^{-1}(f(x)) = \frac{(3x + 5) - 5}{3} = x \)

Thus, both compositions return \( x \), confirming that \( f^{-1}(x) \) is correct.

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Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What is the inverse of \( f(x) = 2x + 7 \)?
2. How do you find the inverse of a quadratic function?
3. What is the significance of the inverse function in real-world applications?
4. How can you graphically verify if two functions are inverses of each other?
5. What is the domain and range of an inverse function?

**Tip:** Always check that a function is one-to-one (bijective) before finding its inverse. If it’s not one-to-one, the inverse doesn't exist over the entire domain!

Learn how to find the inverse function of f(x) = 3x + 5 through a step-by-step solution. This detailed guide explores the concept of inverse functions, covers key algebraic principles, and verifies the solution. Ideal for students in Grades 9-12.

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