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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 \( f(f^{-1}(x)) = x \).

Find the inverse function and graph both f(x) and f^{-1}(x) on the same set of axes. f(x) = 6 - 5x. The inverse function is f^{-1}(x) = ? Choose the correct graph for f(x) and f^{-1}(x) from the given options (A, B, C, D).

Explore the steps to find the inverse function of f(x) = 6 - 5x and graph both f(x) and its inverse f^{-1}(x) on the same set of axes. Learn about the symmetry between a function and its inverse across the line y = x. Suitable for students in Grades 9-12.

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