To solve the given problem, $f(x) = 2x + 9$, and to evaluate $f^{-1}(-1)$ (the value of the inverse function at $-1$):

Steps to Solve:

1. Find the inverse function $f^{-1}(x)$: $f(x) = 2x + 9$ Replace $f(x)$ with $y$: $y = 2x + 9$ Swap $x$ and $y$: $x = 2y + 9$ Solve for $y$ in terms of $x$: $y = \frac{x - 9}{2}$ Thus, the inverse function is: $f^{-1}(x) = \frac{x - 9}{2}$

2. Evaluate $f^{-1}(-1)$: Substitute $-1$ into $f^{-1}(x)$: $f^{-1}(-1) = \frac{-1 - 9}{2} = \frac{-10}{2} = -5$

Final Answer:

$f^{-1}(-1) = -5$

Would you like a step-by-step breakdown of any specific part?

Here are 5 related questions to explore further:

1. How do you find the inverse of a quadratic function?
2. What does it mean geometrically for a function and its inverse to reflect over $y = x$?
3. If $g(x) = 3x + 4$, what is $g^{-1}(7)$?
4. How do you determine if a function has an inverse?
5. What are some applications of inverse functions in real-world problems?

Tip: When finding the inverse of a function, ensure the function is one-to-one (passes the horizontal line test).