The problem asks to find the inverse of the function $f(x) = 7x - 9$, and also mentions that the graphs of $f$ and $f^{-1}$ are symmetric with respect to the line $y =$ (a specific value needs to be filled in).

Part (a): Finding the inverse function

Given $f(x) = 7x - 9$, to find the inverse, follow these steps:

1. Replace $f(x)$ with $y$, so the equation becomes: $y = 7x - 9$
2. Swap $x$ and $y$ to begin finding the inverse: $x = 7y - 9$
3. Solve for $y$: $x + 9 = 7y$ $y = \frac{x + 9}{7}$
4. Therefore, the inverse function is: $f^{-1}(x) = \frac{x + 9}{7}$

Part (b): Symmetry of graphs

The graphs of a function and its inverse are always symmetric with respect to the line $y = x$. Therefore, the answer for this part is: $y = x$

To summarize:

• Part (a): $f^{-1}(x) = \frac{x + 9}{7}$
• Part (b): $y = x$

Do you need any further details or clarifications?

Here are five related questions to help deepen your understanding:

1. How do you verify if two functions are inverses of each other?
2. Why are the graphs of a function and its inverse symmetric with respect to $y = x$?
3. Can every function have an inverse? What conditions are necessary for a function to be invertible?
4. How would you graphically represent both $f(x) = 7x - 9$ and $f^{-1}(x)$ on the same coordinate plane?
5. How does the derivative of a function relate to the derivative of its inverse?

Tip: When finding the inverse of a function, swapping $x$ and $y$ is key, but don’t forget to check if the original function is one-to-one to ensure it has an inverse!