# Find Range of Absolute Value Functions

Find the range of absolute value functions; examples and matched problems with their answers at the bottom of this page are included.

### 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 ( -∞ , +∞)
As x takes values from -∞ to +∞, |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| ≥ 0 (see graphs below)

In interval form, the range of y = |ax + b| is given by [0 , +∞) or by the inequality y ≥ 0

#### 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| ≥ 0
• Multiply the two sides of the above inequality by -1 and change the symbol of inequality to obtain
- |x| ≤ 0
• Hence the range of -|x| is also given by the interval
(-∞ , 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| ≥ 0
• Multiply the two sides of the inequality by 2 to write
2 |2x + 4| ≥ 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 , ∞)

#### 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| ≥ 0
• Multiply both sides of the inequality by 2 to obtain
2 |-4x + 5| ≥ 0

• Add -4 to both sides of the above inequality to obtain
2 |-4x + 5| - 4 ≥ - 4
• The range of 2 |-4x + 5| - 4 may also be written in interval form as follows
[-4 , ∞)

#### 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| ≥0
• Multiply all terms of the inequality by -(1/4) and change the symbol of inequality to obtain
-(1/4) |-4x + 5| ≤0

• Add 1/2 to both sides of the above inequality to obtain
-(1/4) |-4x + 5| + 1/2 ≤ 1/2
• The range of values of -(1/4) |-4x + 5| + 1/2 may also be written in interval form as follows
(-∞ , 1/2]

#### Matched Problem 4:

Find the range of function f defined by
f(x) = 4|-6x - 1/2| - 5

1)   (-∞ , 0]

2)   (-∞ , 0]

3)   (-∞ , -6]

4)   [-5 , +∞)