Evaluating

To find $f(a)$:

1. Determine which condition $a$ satisfies
2. Use the corresponding formula

Example: For the function above:

• $f(-2) = -2 + 1 = -1$ (use $x + 1$ since $-2 < 0$)
• $f(3) = 3^2 = 9$ (use $x^2$ since $3 \geq 0$)

Graphing

1. Graph each piece on its domain
2. Use open circles for endpoints NOT included (< or >)
3. Use closed circles for endpoints included (≤ or ≥)

Common Types

Absolute Value: $|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

Step Functions: Different constant values on intervals

Continuity

Check if pieces "connect" at boundary points.

Continuous if: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$

From right: (use $3x - 1$) $\lim_{x \to 1^+} f(x) = 3(1) - 1 = 2$

Function value: (use $x^2 + 1$ since $1 \leq 1$) $f(1) = 1^2 + 1 = 2$

Since all three equal 2, the function is continuous at $x = 1$.

Answer: Yes, continuous at $x = 1$