CPM Homework Help : CALC3RD Problem 8-31

Determine the value of $h$ so that each of the functions below is continuous.

$f ( x ) = \left\{ \begin{array} { c l } { h } & { \text { for } x = 3 } \\ { \frac { x ^ { 2 } - x - 6 } { x - 3 } } & { \text { for } x \neq 3 } \end{array} \right.$
Hint (a):
After factoring, you will see that the bottom piece has a hole at $x = 3$ . If the function is to be continuous, then the hole needs to be filled in.
What $y$ -value will fill in the hole?

$f ( x ) = \left\{ \begin{array} { c c } { \sqrt { x + h } } & { \text { for } x \leq 3 } \\ { - x + h } & { \text { for } x > 3 } \end{array} \right.$
Step 1(b):
$\sqrt{3+h}=-3+h$
Hint (b):
Side note: Even after determining the value of $h$ that makes $y = f(x)$ continuous at $x = 3$ , $f(x)$ will remain discontinuous at $x = 0$ .

Determine where the functions in parts (a) and (b) are differentiable.
Hint (c):
Does the derivative exist at the boundary points? In other words, do the slopes agree from the left and the right?