Domain and Range of Absolute Value Functions I

Consider the absolute value function definition given below and the associated graph.

$\left|3x+2\right|$|3x+2|$=$=		$3x+2$3x+2$,$,	$x\ge-\frac{2}{3}$x≥−23​
		$-\left(3x+2\right)$−(3x+2)$,$,	$x<-\frac{2}{3}$x<−23​

 

The domain of this function is the whole real line - any value of $x$x can be substituted in, large or small, positive or negative. On the other hand, the range of the absolute value function illustrated above is the interval $[0,\infty)$[0,∞). Any non-negative value is a possible output, but we cannot obtain a negative output from any $x$x-value.

Also note that $|-3x-2|$|−3x−2| is just another way to define the same function. For a given value of $x$x the function $-3x-2$−3x−2 (without absolute value signs) has almost the same output as the function $3x+2$3x+2 - it just differs by its sign. The absolute value around each of these functions then makes the outputs exactly the same.

This is true in general: For all values of $a$a and $b$b (each positive or negative), 

$|-ax-b|=|(-1)\times(ax+b)|=|-1|\times|ax+b|=|ax+b|$|−ax−b|=|(−1)×(ax+b)|=|−1|×|ax+b|=|ax+b|.

This trick can be convenient whenever we want to move minus signs around inside the absolute value.

Worked example

Solve: Describe the natural domain and the range of the function given by the expression $|-x+2|$|−x+2|.

Think: The absolute value function can take any number into it - no values of $x$x can break the function. What kinds of numbers $k$k could be the output? In mathematical terms, for what values of $k$k can $k=|-x+2|$k=|−x+2|? Think about what value of $x$x makes $k=0$k=0. We could also use the trick discussed above and discuss the expression $|x-2|$|x−2|, since it is equal to $|-x+2|$|−x+2|, but it is not necessary.

Do: The domain of the function is the entire real line, $\left(-\infty,\infty\right)$(−∞,∞).

As for the range, we can tell that $x=2$x=2 produces $k=0$k=0, and any value of $x$x either less than or greater than $2$2 will produce a positive value of $k$k. To confirm this, let's consider each case separately.

If $x>2$x>2 then $-x+2$−x+2 will be negative. This means 

$k=-(-x+2)=x-2$k=−(−x+2)=x−2,

and by substituting a large enough $x$x we can make $k>0$k>0 as large as we like.

If $x<2$x<2 then $-x+2$−x+2 will be positive. This means $k=-x+2$k=−x+2, and by substituting a large enough $x$x we can make $k>0$k>0 as large as we like once again.

Taking these cases together we conclude that $k\ge0$k≥0 is the range. We have confirmed in algebraic symbols what is evident from a graph of the function: the range is the non-negative real numbers.

Reflect: Do all absolute value functions have domain $\left(-\infty,\infty\right)$(−∞,∞) and range $\left[0,\infty\right)$[0,∞)? When might this not be the case?

Practice questions

Question 1

Question 2

Question 3