### 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.
So I'll start by rewriting f(x) as y:
\[y = \frac{2x + 5}{x - 11}.\]
Now I'll swap x and y:
\[x = \frac{2y + 5}{y - 11}.\]
To solve for y, I'll first cross-multiply to get rid of the fraction:
\[x(y - 11) = 2y + 5.\]
Then I'll expand the left side and move all the y terms to one side:
\[xy - 11x = 2y + 5 \implies xy - 2y = 11x + 5.\]
I can factor out y from the left side:
\[y(x - 2) = 11x + 5.\]
And finally, I can divide both sides by x - 2 to isolate y:
\[y = \frac{11x + 5}{x - 2}.\]
This is the inverse function of f, which I can write as f^{-1}(x):
\[f^{-1}(x) = \frac{11x + 5}{x - 2}.\]