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Wize AP Calculus (AB) Textbook > Review: Functions/Pre-Calculus

Absolute Value Functions

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The Absolute Value Function

The absolute value function is defined as follows:

∣x∣={xif x≥0−xif x<0\boxed{|x| = \begin{cases} x\hspace{1.06cm} \text{if } x\geq0\\ -x\hspace{0.8cm}\text{if } x<0\\ \end{cases}}



More generally, if we would like to consider the composition of the absolute value function with another function:

∣f(x)∣={f(x)for x such that f(x)≥0−f(x)for x such that f(x)<0\boxed{|f(x)| = \begin{cases} f(x)\hspace{1.06cm} \text{for x such that } f(x)\geq0\\ -f(x)\hspace{0.8cm} \text{for x such that } f(x)<0\\ \end{cases} }

In these cases, we must determine the intervals on which our function is positive and negative respectively.




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Example: Absolute Value Functions

Write out a piece-wise definition for the function

f(x)=∣3x2−1∣f(x) = |3x^2-1|

First we should consider when 3x2−13x^2-1 is positive.
3x2−1≥03x2≥1x2≥13x≤−13 or x≥13\begin{aligned} & 3x^2-1 \geq 0 \\ & 3x^2 \geq 1 \\ & x^2 \geq \frac{1}{3} \\ & x \leq \frac{-1 }{\sqrt{3}} \text{ or } x \geq \frac{1 }{\sqrt{3}} \end{aligned}
Next, we can consider when 3x2−13x^2-1 is negative.
3x2−1<03x2<1x2<13−13<x<13\begin{aligned} & 3x^2-1 < 0 \\ & 3x^2 < 1 \\ & x^2 < \frac{1}{3} \\ & \frac{-1 }{\sqrt{3}} < x < \frac{1 }{\sqrt{3}} \end{aligned}
Finally, we can determine the piecewise definition of f(x)f(x).
f(x)=∣3x2−1∣={3x2−1 if x≤−13 or x≥13−3x2+1 if −13<x<13f(x) = |3x^2-1| = \begin{cases} 3x^2-1 \hspace{1.0 cm}\text{ if } x \leq \frac{-1 }{\sqrt{3}} \text{ or } x \geq \frac{1 }{\sqrt{3}} \\ -3x^2 + 1 \hspace{0.74 cm}\text{ if } \frac{-1 }{\sqrt{3}} < x < \frac{1 }{\sqrt{3}} \end{cases}