Bridging Two Broken Ends

A function is assembled from three pieces, with constants $a$ and $b$ still to be chosen:

$f(x) = \begin{cases} \dfrac{x^2 - x - 12}{x - 4}, & x < 4 \\ ax + b, & 4 \le x \le 9 \\ \dfrac{x - 9}{\sqrt{x} - 3}, & x > 9 \end{cases}$

The middle piece is a straight bridge laid between the two outer pieces. Note that each outer formula is undefined at the joint it runs up to, so only the middle piece supplies an actual value at $x = 4$ and at $x = 9$.

There is exactly one pair $(a, b)$ that makes $f$ continuous at every real number — no holes, no jumps, anywhere.

With that pair in place, report $f(5)$, rounded to two decimal places.