# Piecewise Functions: Definition, Absolute Value, and Graphing Techniques **Core Concept:** Piecewise functions use different formulas over different intervals, and can be graphed by plotting each piece within its designated domain using open and closed circles at boundaries. --- ## Definition A piecewise function is one where the formula changes depending on the value of x. The function is defined by different expressions over different intervals of the domain. ## The Absolute Value Function The simplest piecewise function is absolute value: f(x) = |x| Absolute value gives the distance from zero. Practically: - If x is positive, leave it alone - If x is negative, change its sign **Piecewise definition**: $$f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ For example: |-12| = -(-12) = 12. The formula "-x" changes the sign of a negative input to produce a positive output. **Graph**: The absolute value function forms a V-shape, with vertex at the origin. ## Graphing Piecewise Functions **Method**: Graph each piece individually over its specified domain, then combine. ### Example: Graphing |x| 1. **For x ≥ 0**: Graph y = x (a line through origin with slope 1), but only for x ≥ 0. This gives the right half of the V. 2. **For x < 0**: Graph y = -x (a line through origin with slope -1), but only for x < 0. This gives the left half of the V. Use a **closed circle** (●) where the inequality includes equality (≤ or ≥), and an **open circle** (○) where it doesn't (< or >). ### Example: A Three-Piece Function $$f(x) = \begin{cases} 0 & \text{if } x \leq -1 \\ \sqrt{1-x^2} & \text{if } -1 < x < 1 \\ x & \text{if } x \geq 1 \end{cases}$$ **Step 1**: Identify transition points: x = -1 and x = 1 **Step 2**: Graph each piece: - **For x ≤ -1**: y = 0 is a horizontal line along the x-axis. Draw from x = -1 leftward, with a closed circle at (-1, 0). - **For -1 < x < 1**: y = √(1 - x²). Squaring both sides: y² = 1 - x², which rearranges to x² + y² = 1. This is a circle centered at the origin with radius 1. Since y = √(1 - x²) (positive square root only), this is the **top half** of the circle. Draw the semicircle from x = -1 to x = 1, with open circles at both endpoints. - **For x ≥ 1**: y = x is a diagonal line with slope 1. Draw from x = 1 rightward, with a closed circle at (1, 1). **Important**: The inequalities must not overlap at the same x-value with different function values, or the result wouldn't be a function. ## Graphing Piecewise Functions (Summary) A piecewise function is defined by different expressions over different intervals of the x-axis. To graph one: 1. **Identify the intervals** on the x-axis where each piece applies 2. **Graph each piece** only within its designated interval 3. **Mark endpoints carefully** — use closed circles (●) when the inequality includes "equals" (≤ or ≥) and open circles (○) when it doesn't (< or >) When pieces meet at boundary points, one function's endpoint may be open while another's is closed, ensuring no overlapping points and maintaining the definition of a function. --- **Key Takeaways:** - Piecewise functions use different formulas for different x-intervals, with absolute value being the simplest example - Graph each piece only within its designated interval, using closed circles for ≤/≥ and open circles for at boundaries - The pieces must not overlap at the same x-value with different outputs, or it wouldn't be a function