### 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 swap the roles of x and y and then solve for y.
So, I'll start by writing
\[y = \frac{2x + 5}{x - 11}\]
and then swap x and y to get
\[x = \frac{2y + 5}{y - 11}.\]
Now, I need to solve for y in terms of x. I'll do this by cross-multiplying and then rearranging the terms.
So, I have
\[(x)(y - 11) = (2y + 5)\]
which expands to
\[xy - 11x = 2y + 5.\]
To isolate y, I'll subtract xy and 5 from both sides, and then factor out y. This gives me
\[-11x - 5 = y(2 - x)\]
or
\[y = \frac{-11x - 5}{2 - x}.\]
This is the inverse function of f(x), which I can write as
\[f^{-1}(x) = \frac{-11x - 5}{2 - x}.\]