## Introduction to Absolute Value Functions

### LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

• Graph an absolute value function.
• Solve an absolute value equation.
• Solve an absolute value inequality.

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate absolute value functions.

## Understanding Absolute Value

Recall that in its basic form $\displaystyle{f}\left({x}\right)={|x|}$, 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

$f(x) = \begin{cases} x ,\ x \geq 0 \\ -x , x < 0\\ \end{cases}$

### Example 1: Determine a Number within a Prescribed Distance

Describe all values $x$ within or including a distance of 4 from the number 5.

### Solution

Figure 2

We want the distance between $x$ 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 $x$ to 5 can be represented using the absolute value as $|x - 5|$. We want the values of $x$ that satisfy the condition $|x - 5|\le 4$.

### Analysis of the Solution

Note that

$\displaystyle{-4}\le{x - 5}$
$\displaystyle{1}\le{x}$
And:
$\displaystyle{x-5}\le{4}$
$\displaystyle{x}\le{9}$

So $|x - 5|\le 4$ is equivalent to $1\le x\le 9$.

However, mathematicians generally prefer absolute value notation.

### Try It 1

Describe all values $x$ within a distance of 3 from the number 2.

Solution

### Example 2: Resistance of a Resistor

Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often $\displaystyle\text{\pm 1%,}\pm\text{5%,}$ or $\displaystyle\pm\text{10%}$.

Suppose we have a resistor rated at 680 ohms, $\pm 5%$. Use the absolute value function to express the range of possible values of the actual resistance.

### Solution

5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance $R$ in ohms,

$|R - 680|\le 34$

### Try It 2

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

Solution