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.