Piecewise Behavior Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Write ∣x∣|x| as an explicit piecewise function, evaluate ∣−4∣|-4|, ∣0∣|0|, and ∣7∣|7|, and sketch its graph.

Solution

  1. 1
    Definition: ∣x∣={xif x≥0−xif x<0|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}.
  2. 2
    Evaluate: ∣−4∣=−(−4)=4|-4| = -(-4) = 4; ∣0∣=0|0| = 0; ∣7∣=7|7| = 7.
  3. 3
    Graph: two rays meeting at the origin (0,0)(0,0), slope −1-1 for x<0x<0 and slope +1+1 for x≥0x\geq0, forming a 'V' shape.

Answer

∣−4∣=4|-4|=4, ∣0∣=0|0|=0, ∣7∣=7|7|=7; V-shaped graph with vertex at origin
The absolute value function is the simplest and most important piecewise function. It measures distance from zero, always returning a non-negative value. Its V-shape has slope ±1\pm1 and vertex at the origin.

About Piecewise Behavior

Piecewise behavior refers to a function that exhibits qualitatively different characteristics in different regions of its domain, like having a different slope or curvature in each region.

Learn more about Piecewise Behavior →

More Piecewise Behavior Examples

Example 2 medium

Solve the equation [formula] and the inequality [formula].

Example 3 easy

Simplify [formula] for [formula], [formula], and [formula].

Example 4 hard

Express [formula] as a piecewise function (without absolute value bars) and identify where [formula]

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