### Instruction: Find the inverse of the function
\[f(x) = \frac{2x + 5}{x - 11}.\].### Response: To find the inverse of a function, I need to switch the roles of x and y and solve for y.
So, I start by writing the function as y = f(x) and then swapping x and y: \[x = \frac{2y + 5}{y - 11}.\]
Now, I want to isolate y on one side of the equation.
To do that, I can cross-multiply and get: \[xy - 11x = 2y + 5.\]
Then, I can move all the terms with y to one side and all the terms without y to the other side: \[xy - 2y = 11x + 5.\]
I can factor out y from the left-hand side: \[y(x - 2) = 11x + 5.\]
And then divide both sides by (x - 2) to get y by itself: \[y = \frac{11x + 5}{x - 2}.\]
This is the inverse function of f(x).