### 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 then solve for y in terms of x.
So, I start by writing
\[y = \frac{2x + 5}{x - 11}\]
and then swap x and y to get
\[x = \frac{2y + 5}{y - 11}.\]
Next, I need to isolate y on one side of the equation.
I can do this by cross-multiplying and then rearranging the terms.
I get
\[xy - 11x = 2y + 5\]
and then
\[xy - 2y = 11x + 5.\]
To make y the subject, I can factor out y from the left-hand side and then divide both sides by the coefficient of y.
I get
\[y(x - 2) = 11x + 5\]
and then
\[y = \frac{11x + 5}{x - 2}.\]
This is the inverse function of f(x).