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To find the inverse of the given functions, we'll follow the standard steps. Let's handle each function separately:

### 1. Inverse of \( m(x) = 7x - 5 \)

**Step-by-Step Process**:
1. **Replace** \( m(x) \) with \( y \):  
   \( y = 7x - 5 \)
   
2. **Swap** \( x \) and \( y \) (this is the key step to finding the inverse):  
   \( x = 7y - 5 \)
   
3. **Solve** for \( y \):  
   Add 5 to both sides:  
   \( x + 5 = 7y \)  
   Now divide both sides by 7:  
   \( y = \frac{x + 5}{7} \)
   
4. **Replace** \( y \) with \( m^{-1}(x) \) to denote the inverse function:  
   \( m^{-1}(x) = \frac{x + 5}{7} \)

So, the inverse of \( m(x) = 7x - 5 \) is \( m^{-1}(x) = \frac{x + 5}{7} \).

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### 2. Inverse of \( f(x) = 3x - 8 \)

**Step-by-Step Process**:
1. **Replace** \( f(x) \) with \( y \):  
   \( y = 3x - 8 \)
   
2. **Swap** \( x \) and \( y \):  
   \( x = 3y - 8 \)
   
3. **Solve** for \( y \):  
   Add 8 to both sides:  
   \( x + 8 = 3y \)  
   Now divide both sides by 3:  
   \( y = \frac{x + 8}{3} \)
   
4. **Replace** \( y \) with \( f^{-1}(x) \) to denote the inverse function:  
   \( f^{-1}(x) = \frac{x + 8}{3} \)

So, the inverse of \( f(x) = 3x - 8 \) is \( f^{-1}(x) = \frac{x + 8}{3} \).

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Would you like a detailed breakdown of any of these steps? Here are five follow-up questions:

1. What are the four steps involved in finding the inverse of a function?
2. How can you verify that two functions are inverses of each other?
3. Why do we swap \( x \) and \( y \) when finding the inverse?
4. How does the slope of the original function compare with its inverse?
5. What is the general form of an inverse for a linear function \( f(x) = ax + b \)?

**Tip**: Always remember to check your work by plugging the inverse function back into the original function. If \( f(f^{-1}(x)) = x \), then you have found the correct inverse.

Find the inverse of the following functions: 1. m(x) = 7x - 5, 2. f(x) = 3x - 8

This guide explores the step-by-step process to find the inverse of linear functions m(x) = 7x - 5 and f(x) = 3x - 8. Learn how to switch variables and solve for y, using basic algebra. This lesson is ideal for high school students learning about function inverses.

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