# 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

**absolute value functions**.

## 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]

**Q & A**

**Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?**

*Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.*

*No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points.*

## 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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