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2-14.

Given the function $f ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } - 1 } & { \text { for } x < 2 } \\ { 2 x + 3 } & { \text { for } x \geq 2 } \end{array} \right.$ , evaluate the following:

$f(2)$
Hint (a):
Notice the inequality symbols. $f(2)$ exists as part of the right side of $f(x)$ not the left.
Answer (a):
$f(2) = 2(2)+3 = 7$

As $x → 2^+$ , $y →$ ?
Hint (b):
Compare this question to part (a).

As $x → 2^-$ , $y →$ ?
Hint (c):
Compare this question to part (a). Even though $f(2)$ does not exist in the left side of the equation, since both sides approach $x = 2$ we can evaluate what it approaches.
Answer (c):
$x →2^−$ , $y →(2)^²−1$ or $y →3$

What do the results from parts (b) and (c) indicate about the graph?
Hint (d):
Is $f(x)$ continuous at $x = 2$ ? In other words, does $f(x)$ approach the same $y$ -value from the left and from the right of $x = 2$ ?

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