Piecewise Behavior Math Example 4

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

Example 4

hard
Express f(x)=∣x2−4∣f(x) = |x^2 - 4| as a piecewise function (without absolute value bars) and identify where f(x)=0f(x) = 0.

Solution

  1. 1
    Find where x2−4x^2-4 changes sign: x2−4=0⇒x=±2x^2-4=0 \Rightarrow x=\pm 2. For ∣x∣>2|x|>2: x2>4x^2>4 so x2−4>0x^2-4>0. For ∣x∣<2|x|<2: x2<4x^2<4 so x2−4<0x^2-4<0.
  2. 2
    f(x)={x2−4∣x∣≥2−(x2−4)=4−x2∣x∣<2f(x) = \begin{cases} x^2-4 & |x| \geq 2 \\ -(x^2-4) = 4-x^2 & |x| < 2 \end{cases}
  3. 3
    Zeros: f(x)=0f(x)=0 when x2−4=0⇒x=±2x^2-4=0 \Rightarrow x = \pm 2.

Answer

f(x)={x2−4∣x∣≥24−x2∣x∣<2f(x) = \begin{cases} x^2-4 & |x|\geq2\\ 4-x^2 & |x|<2\end{cases}; zeros at x=±2x=\pm2
To remove absolute value bars algebraically, determine where the expression inside is positive or negative, then write separate rules. The zeros of ∣g(x)∣|g(x)| occur exactly at the zeros of g(x)g(x).

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.

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More Piecewise Behavior Examples

Example 1 easy

Write [formula] as an explicit piecewise function, evaluate [formula], [formula], and [formula], and

Example 2 medium

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

Example 3 easy

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

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