Absolute value functions

Absolute values turn negative values into positive ones, and leave positive ones unchanged. Absolute value functions are useful for expressing situations where only magnitude matters, without regard to direction, like measuring distance, temperature differences, profit or loss diffeerences, and margin of error.

Absolute values

The absolute value of any number $x$ gives a measure of its size or magnitude, regardless of whether it is positive or negative. It is the distance from $0$ on a number line.

We represent the absolute value of $x$ using two vertical lines $\left|\right|$. For example, the absolute value of $x$ would be $\left|x\right|$.

Example 1 – finding the absolute value

Find $\left|2\right|$.

To find the absolute value, we simply turn any negative value into a positive one. $2$ is already positive, so $\left|2\right|=2$.

Exercise – finding the absolute value

Evaluate the following.

1. $|-11|$
2. $|-9+4|$
3. $-\left|4\right|-|-5|$
4. $|-12|-\left|3\right|$
5. $|-30|÷\left|5\right|$

Solving equations involving absolute values

Equations involving absolute values often have two solutions because the absolute value measures distance from $0$ and is always positive. This means the expression inside can be equal to either a positive or negative value.

For example, if $\left|x\right|=7$, $x$ can be $7$ or $-7$ since both give an answer of $7$.

Example 1 – solving equations involving absolute values

Solve $|x-2|=3$.

We consider two cases: $x-2=3$ and $x-2=-3$.

For $x-2=3$:
$$\begin{array}{rl}x-2& =3\\ x& =5\end{array}$$

For $x-2=-3$:
$$\begin{array}{r}x-2=-3\\ x& =-1\end{array}$$

Therefore, $x=-1$ and $x=5$.

Exercise – solving equations involving absolute values

Find the following, for $x\in \mathbb{R}$.

1. $\{x:\left|x\right|=6\}$
2. $\{x:|x-1|<3\}$
3. $\{x:\left|{\displaystyle \frac{x-3}{2}}\right|\ge 1\}$
4. $\{x:\left|{\displaystyle \frac{x}{2}}\right|=|x+2|\}$

Graphs of absolute value functions

Absolute value functions are hybrid functions. They have the general form:

$$f:\mathbb{R}\to \mathbb{R},\text{where }f(x)=\left|x\right|=\{\begin{array}{ll}-x& x<0\\ x& x\ge 0\end{array}$$

The graph of $y=\left|x\right|$ is shown.

• The domain of $y=\left|x\right|$ is $\mathbb{R}$, the set of all real numbers.
• The range of $y=\left|x\right|$ is ${\mathbb{R}}^{+}\cup \left\{0\right\}$, the set of all positive real numbers and $0$.

In general, to sketch the graph of $y=|f(x)|$, we need to sketch the graph of $y=f(x)$ first, then reflect the $x$-axis portion of the graph below the $x$-axis.

Transforming graphs of absolute value functions

Absolute value functions can be transformed like other graphs and functions. Let's look at the basic form $y=\left|x\right|$ as an example.

Example 1 – transforming graphs of absolute value functions

Graph $y=|x-2|$.

The $x-2$ indicates that the graph is being translated $2$ units to the right.

Exercise – transforming graphs of absolute value functions

Sketch the following graphs.

1. $y=|x+4|$
2. $y=|x-1|-3$
3. $y=|3-x^2|$

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