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

8-31.

Find h so that the functions below are 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 $f(x)$ were continuous, then that hole needs to be filled in.
What $y$-value would fill in the hole?

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

Side note: Even after finding the value of $h$ that makes $f(x)$ 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?