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get the inverse of the following problems and give me the solutions:

Let's find the inverse functions for the given problems. The steps for finding the inverse involve swapping \( x \) and \( y \) (where \( y = f(x) \)) and solving for \( y \).

### 1. \( f(x) = \frac{1}{4}x - 3 \)

#### Step-by-Step Solution:
1. Write \( y = f(x) \):
   \[
   y = \frac{1}{4}x - 3
   \]
2. Swap \( x \) and \( y \):
   \[
   x = \frac{1}{4}y - 3
   \]
3. Solve for \( y \):
   \[
   x + 3 = \frac{1}{4}y
   \]
   \[
   y = 4(x + 3)
   \]
   \[
   y = 4x + 12
   \]
4. Therefore, the inverse function is:
   \[
   f^{-1}(x) = 4x + 12
   \]

### 2. \( f(x) = \frac{8}{x - 5} \)

#### Step-by-Step Solution:
1. Write \( y = f(x) \):
   \[
   y = \frac{8}{x - 5}
   \]
2. Swap \( x \) and \( y \):
   \[
   x = \frac{8}{y - 5}
   \]
3. Solve for \( y \):
   \[
   x(y - 5) = 8
   \]
   \[
   xy - 5x = 8
   \]
   \[
   xy = 8 + 5x
   \]
   \[
   y = \frac{8 + 5x}{x}
   \]
4. Therefore, the inverse function is:
   \[
   f^{-1}(x) = \frac{8 + 5x}{x}
   \]

### 3. \( f(x) = \frac{7x + 2}{x + 9} \)

#### Step-by-Step Solution:
1. Write \( y = f(x) \):
   \[
   y = \frac{7x + 2}{x + 9}
   \]
2. Swap \( x \) and \( y \):
   \[
   x = \frac{7y + 2}{y + 9}
   \]
3. Solve for \( y \):
   \[
   x(y + 9) = 7y + 2
   \]
   \[
   xy + 9x = 7y + 2
   \]
   \[
   xy - 7y = 2 - 9x
   \]
   \[
   y(x - 7) = 2 - 9x
   \]
   \[
   y = \frac{2 - 9x}{x - 7}
   \]
4. Therefore, the inverse function is:
   \[
   f^{-1}(x) = \frac{2 - 9x}{x - 7}
   \]

### 4. \( g(x) = \frac{7x - 2}{3x + 1} \)

#### Step-by-Step Solution:
1. Write \( y = g(x) \):
   \[
   y = \frac{7x - 2}{3x + 1}
   \]
2. Swap \( x \) and \( y \):
   \[
   x = \frac{7y - 2}{3y + 1}
   \]
3. Solve for \( y \):
   \[
   x(3y + 1) = 7y - 2
   \]
   \[
   3xy + x = 7y - 2
   \]
   \[
   3xy - 7y = -x - 2
   \]
   \[
   y(3x - 7) = -x - 2
   \]
   \[
   y = \frac{-x - 2}{3x - 7}
   \]
4. Therefore, the inverse function is:
   \[
   g^{-1}(x) = \frac{-x - 2}{3x - 7}
   \]

Would you like further details on any of these solutions or have any questions? Here are five related questions you might consider:

1. How do you determine the domain of the inverse function?
2. What happens to the vertical and horizontal asymptotes of the original function in the inverse function?
3. Can all functions have an inverse? Why or why not?
4. How does the concept of one-to-one functions relate to the existence of an inverse?
5. What role does the reflection over the line \( y = x \) play in finding the inverse graphically?

**Tip:** When finding the inverse of a function, always check whether the function is one-to-one, as this ensures that an inverse exists.

Discover how to find inverse functions for various types of functions, including f(x) = (7x + 2) / (x + 9) and f(x) = 8 / (x - 5). Detailed step-by-step solutions provided, covering functions and inverse functions in advanced high school mathematics.

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