Examples and matched problems on how to find the range of absolute value functions are presented along with their answers located at the bottom of this page.
Graphical Analysis of Range of Absolute Value Functions
The range of a function \( y = f(x) \) is the set of values \( y \) takes for all values of \( x \) within the domain of \( f \).
What is the range of \( y = f(x) = |x| \)?
The domain of \( f \) above is the set of all values of \( x \) in the interval \( (-\infty, +\infty) \).
As \( x \) takes values from \( -\infty \) to \( +\infty \), \( |x| \) takes all values from \( 0 \) to infinity and in general absolute value function of the form \( y = |ax + b| \) has a range given by the interval
\(|ax + b| \geq 0\) (see graphs below)
Fig1. - Range of Absolute Value Function.
In interval form, the range of \( y = |ax + b| \) is given by \[ [0 , +\infty) \] or by the inequality \[ y \geq 0 \]
Examples with Solutions on How to Find Range of Absolute Value Functions
Example 1
Find the range of function \( f \) defined by
\[ f(x) = - |x| \]
Solution to Example 1
Start with the range of the basic absolute value function (see discussion above) and write
\[ |x| \geq 0 \]
Multiply the two sides of the above inequality by -1 and change the symbol of inequality to obtain
\[ - |x| \leq 0 \]
Hence the range of \( -|x| \) is also given by the interval
\[ (-\infty , 0] \]
Matched Problem 1
Find the range of function \( f \) defined by
\[ f(x) = - |2x| \]
Example 2
Find the range of function \( f \) defined by
\[ f(x) = 2|2x + 4| \]
Solution to Example 2
The range of \( |2x + 4| \) is given by
\[ |2x + 4| \geq 0 \]
Multiply the two sides of the inequality by 2 to write
\[ 2 |2x + 4| \geq 0 \]
The range of the given function \( f \) is written above in inequality form and may also be written in interval form as follows
\[ [0 , +\infty) \]
Matched Problem 2
Find the range of function \( f \) defined by
\[ f(x) = - 4|2x + 4| \]
Example 3
Find the range of function \( f \) defined by
\[ f(x) = 2|-4x + 5| - 4 \]
Solution to Example 3
The range of \( |-4x + 5| \) is given by
\[ |-4x + 5| \geq 0 \]
Multiply both sides of the inequality by 2 to obtain
\[ 2 |-4x + 5| \geq 0 \]
Add -4 to both sides of the above inequality to obtain
\[ 2 |-4x + 5| - 4 \geq - 4 \]
The range of \( 2 |-4x + 5| - 4 \) may also be written in interval form as follows
\[ [-4 , +\infty) \]
Matched Problem 3
Find the range of function \( f \) defined by
\[ f(x) = - 4|2x + 4| - 6 \]
Example 4
Find the range of function \( f \) defined by
\[ f(x) = - (1/4)|-4x + 5| + 1/2 \]
Solution to Example 4
The range of \( |-4x + 5| \) is given by
\[ |-4x + 5| \geq 0 \]
Multiply all terms of the inequality by \(-(1/4)\) and change the symbol of inequality to obtain
\[ -(1/4) |-4x + 5| \leq 0 \]
Add \(1/2\) to both sides of the above inequality to obtain
\[ -(1/4) |-4x + 5| + 1/2 \leq 1/2 \]
The range of values of \( -(1/4) |-4x + 5| + 1/2 \) may also be written in interval form as follows
\[ (-\infty , 1/2] \]
Matched Problem 4
Find the range of function \( f \) defined by
\[ f(x) = 4|-6x - 1/2| - 5 \]
