Problem
$\begin{array}{l}\text { Let } \\ f(x)=\frac{x+1}{x+9} \\ f^{-1}(-10)=\end{array}$
Solution
Step 1 ：Let the function be \(f(x)=\frac{x+1}{x+9}\).
Step 2 ：To find the inverse of a function, we need to switch the x and y (or f(x)) values. This means we set x = f(y) and solve for y.
Step 3 ：In this case, we need to find the value of x such that \(f(x) = -10\).
Step 4 ：Once we have the value of x, we can substitute it into the inverse function to find the value of \(f^{-1}(-10)\).
Step 5 ：Solving the equation \(\frac{x + 1}{x + 9} = -10\), we find that x = -\frac{91}{11}.
Step 6 ：Therefore, the solution to the equation is \(f^{-1}(-10) = \boxed{-\frac{91}{11}}\).
