### Home > APCALC > Chapter 8 > Lesson 8.1.3 > Problem8-31

8-31.

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

1. $f ( x ) = \left\{ \begin{array} { c l } { h } & { \text { for } x = 3 } \\ { \frac { x ^ { 2 } - x - 6 } { x - 3 } } & { \text { for } x \neq 3 } \end{array} \right.$

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?

1. $f ( x ) = \left\{ \begin{array} { c c } { \sqrt { x + h } } & { \text { for } x \leq 3 } \\ { - x + h } & { \text { for } x > 3 } \end{array} \right.$

$\sqrt{3+h}=-3+h$

Side note: Even after determining the value of $h$ that makes $y = f(x)$ continuous at $x = 3$, $f(x)$ will remain discontinuous at $x = 0$.

1. Determine where the functions in parts (a) and (b) are differentiable.

Does the derivative exist at the boundary points? In other words, do the slopes agree from the left and the right?