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The image shows a problem asking to match the graphs of two functions, \( f(x) = -3x - 1 \) and \( g(x) = 5x + 2 \), with the graphs of their inverses, \( f^{-1}(x) \) and \( g^{-1}(x) \).

We have four graphs displayed in the image, each in a different color: red, blue, black, and green. We are asked to match the colors of these graphs with the corresponding functions and their inverses.

### Step 1: Finding the Inverses

1. **For \( f(x) = -3x - 1 \):**
   To find the inverse, swap \( x \) and \( y \) and solve for \( y \):
   \[
   x = -3y - 1
   \]
   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 \( g(x) = 5x + 2 \):**
   Similarly, swap \( x \) and \( y \) and solve for \( y \):
   \[
   x = 5y + 2
   \]
   Solve for \( y \):
   \[
   x - 2 = 5y
   \]
   \[
   y = \frac{x - 2}{5}
   \]
   Therefore, the inverse function is:
   \[
   g^{-1}(x) = \frac{x - 2}{5}
   \]

### Step 2: Identifying the Graphs

Looking at the colors provided in the image:

- Red, Blue, Black, Green

Given the equations and slopes:
- The slope of \( f(x) = -3x - 1 \) is negative, so its graph will be steep and descending.
- The slope of \( f^{-1}(x) = -\frac{1}{3}x - \frac{1}{3} \) is also negative but less steep.
- The slope of \( g(x) = 5x + 2 \) is positive and steep.
- The slope of \( g^{-1}(x) = \frac{1}{5}x - \frac{2}{5} \) is positive but much less steep.

By visually comparing the slopes and y-intercepts in the graph:
- The red line is the steep positive slope, matching \( g(x) = 5x + 2 \).
- The blue line has a steep negative slope, matching \( f(x) = -3x - 1 \).
- The black line has a shallow positive slope, matching \( g^{-1}(x) = \frac{x - 2}{5} \).
- The green line has a shallow negative slope, matching \( f^{-1}(x) = -\frac{x + 1}{3} \).

### Conclusion

- \( f(x) = -3x - 1 \) matches the **blue** graph.
- \( f^{-1}(x) = -\frac{x + 1}{3} \) matches the **green** graph.
- \( g(x) = 5x + 2 \) matches the **red** graph.
- \( g^{-1}(x) = \frac{x - 2}{5} \) matches the **black** graph.

### Let me know if you'd like more details or further clarification!

Here are five questions that could expand this topic:
1. How do you derive the inverse of a linear function in general?
2. Why do the graphs of a function and its inverse reflect over the line \( y = x \)?
3. How would the inverses of non-linear functions differ from linear functions?
4. Can you explain the importance of the slope and y-intercept when graphing linear functions and their inverses?
5. How can you confirm graphically that two functions are indeed inverses of each other?

#### Tip:
When finding the inverse of a function, always remember to swap the roles of \( x \) and \( y \) and solve for the new \( y \). This method works well for all functions that have inverses.

Match the graphs of f(x) = -3x - 1 and g(x) = 5x + 2 with the graphs of their inverses f^{-1}(x) and g^{-1}(x).

Learn how to match the graphs of the linear functions f(x) = -3x - 1 and g(x) = 5x + 2 with their inverses. This problem covers key mathematical concepts, including inverse functions, graphing, and linear equations. Suitable for students in Grades 9-12.

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