We are given the function $f(x) = 6 - 5x$ and asked to find its inverse function $f^{-1}(x)$ and choose the correct graph that shows both the original and inverse functions.

Step 1: Find the Inverse Function

To find the inverse function, we follow these steps:

1. Start with $f(x) = y = 6 - 5x$.
2. Swap $x$ and $y$ to get $x = 6 - 5y$.
3. Solve for $y$ in terms of $x$:

$x = 6 - 5y$ $5y = 6 - x$ $y = \frac{6 - x}{5}$

Thus, the inverse function is: $f^{-1}(x) = \frac{6 - x}{5}$

Step 2: Check the Graphs

The graphs of $f(x)$ and $f^{-1}(x)$ are symmetric across the line $y = x$. We need to identify the correct pair of graphs from the options (A, B, C, D) based on this symmetry.

• The blue line represents $f(x) = 6 - 5x$, which is a line with a slope of $-5$.
• The red dashed line represents $f^{-1}(x) = \frac{6 - x}{5}$, which has a slope of $-\frac{1}{5}$.

Looking at the graphs:

• In the correct graph, the blue line should have a steep negative slope, while the red dashed line should have a shallow negative slope. The graphs should also reflect each other across the line $y = x$.

Correct Answer: Option C

The graph in option C shows the required symmetry, with the original function (solid blue line) and its inverse (dashed red line) reflecting each other over the line $y = x$.

Would you like more details or have any questions?

Related Questions:

1. What is the general method to find the inverse of a linear function?
2. How can we verify if two functions are inverses of each other using composition?
3. Why is the line $y = x$ important when graphing a function and its inverse?
4. How do you graph the inverse of a function using its equation?
5. How does the slope of a function relate to the slope of its inverse?

Tip: When checking if two functions are inverses, you can substitute one into the other and see if the result is the identity function $f(f^{-1}(x)) = x$.