https://file.aiquickdraw.com/mathbot/file/1726529774868xna2c7dg.jpg

Let's extract the relevant information from the image to address the inverse function problems. Here's a summary of the tasks from Section C in the image:

**C. Find the inverse of the given function:**

1. \( f(x) = 3x - 1 \)
2. \( g(x) = \frac{1}{2}x + 4 \)

### Finding the Inverse of Each Function

1. **For \( f(x) = 3x - 1 \):**
   - To find the inverse, replace \( f(x) \) with \( y \):
     \[ y = 3x - 1. \]
   - Swap \( x \) and \( y \):
     \[ x = 3y - 1. \]
   - Solve for \( y \):
     \[ x + 1 = 3y, \]
     \[ y = \frac{x + 1}{3}. \]
   - Therefore, the inverse function is:
     \[ f^{-1}(x) = \frac{x + 1}{3}. \]

2. **For \( g(x) = \frac{1}{2}x + 4 \):**
   - Replace \( g(x) \) with \( y \):
     \[ y = \frac{1}{2}x + 4. \]
   - Swap \( x \) and \( y \):
     \[ x = \frac{1}{2}y + 4. \]
   - Solve for \( y \):
     \[ x - 4 = \frac{1}{2}y, \]
     \[ y = 2(x - 4). \]
   - Therefore, the inverse function is:
     \[ g^{-1}(x) = 2(x - 4). \]

Would you like more details or explanations for these steps?

### Related Questions to Further Your Understanding
1. What is the general process for finding the inverse of a function?
2. How do you determine if a function has an inverse that is also a function?
3. Can a function have more than one inverse?
4. What are the characteristics of a one-to-one function?
5. How can you verify that your inverse function is correct?

**Tip:** A function has an inverse if and only if it is one-to-one (passes both the horizontal and vertical line tests).

Learn how to find the inverse of functions such as f(x) = 3x - 1 and g(x) = 1/2x + 4. This algebra problem involves key concepts in inverse functions, with detailed solutions provided. Suitable for Grades 10-12.

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