Absolute Value Functions
Learning Objectives
By the end of this lesson, you will be able to:- Graph an absolute value function.
- Find the intercepts of an absolute value function
Understanding Absolute Value
Recall that in its basic form [latex]\displaystyle{f}\left({x}\right)={|x|}[/latex], the absolute value function, is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.A General Note: Absolute Value Function
The absolute value function can be defined as a piecewise function$latex f(x) = \begin{array}{l} x ,\ x \geq 0 \\ -x , x < 0\\ \end{array} $
Example: Determine a Number within a Prescribed Distance
Describe all values [latex]x[/latex] within or including a distance of 4 from the number 5.Answer: We want the distance between [latex]x[/latex] and 5 to be less than or equal to 4. We can draw a number line to represent the condition to be satisfied. The distance from [latex]x[/latex] to 5 can be represented using the absolute value as [latex]|x - 5|[/latex]. We want the values of [latex]x[/latex] that satisfy the condition [latex]|x - 5|\le 4[/latex].
Analysis of the Solution
Note that [latex-display]\displaystyle{-4}\le{x - 5}[/latex-display] [latex-display]\displaystyle{1}\le{x}[/latex-display] And: [latex-display]\displaystyle{x-5}\le{4}[/latex-display] [latex-display]\displaystyle{x}\le{9}[/latex-display] So [latex]|x - 5|\le 4[/latex] is equivalent to [latex]1\le x\le 9[/latex]. However, mathematicians generally prefer absolute value notation.Try It
Describe all values [latex]x[/latex] within a distance of 3 from the number 2.Answer: [latex-display]|x - 2|\le 3[/latex-display]
Find the Intercepts of an Absolute Value Function
Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.How To: Given the formula for an absolute value function, find the horizontal intercepts of its graph.
- Isolate the absolute value term.
- Use [latex]|A|=B[/latex] to write [latex]A=B[/latex] or [latex]\mathrm{-A}=B[/latex], assuming [latex]B>0[/latex].
- Solve for [latex]x[/latex].
Example: Finding the Zeros of an Absolute Value Function
For the function [latex]f\left(x\right)=|4x+1|-7[/latex] , find the values of [latex]x[/latex] such that [latex]\text{ }f\left(x\right)=0[/latex] .Answer:
[latex]\begin{array}{l}0=|4x+1|-7\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \text{Substitute 0 for }f\left(x\right).\hfill \\ 7=|4x+1|\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \text{Isolate the absolute value on one side of the equation}.\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ 7=4x+1\hfill & \text{or}\hfill & \hfill & \hfill & \hfill & -7=4x+1\hfill & \text{Break into two separate equations and solve}.\hfill \\ 6=4x\hfill & \hfill & \hfill & \hfill & \hfill & -8=4x\hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ x=\frac{6}{4}=1.5\hfill & \hfill & \hfill & \hfill & \hfill & \text{ }x=\frac{-8}{4}=-2\hfill & \hfill \end{array}[/latex]
The function outputs 0 when [latex]x=1.5[/latex] or [latex]x=-2[/latex].Try It
For the function [latex]f\left(x\right)=|2x - 1|-3[/latex], find the values of [latex]x[/latex] such that [latex]f\left(x\right)=0[/latex].Answer: [latex]x=-1[/latex] or [latex]x=2[/latex]
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