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)
Range of Absolute Value Function
Fig1. - Range of Absolute Value Function.

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


Answers for Matched Problems

1)   (-∞ , 0]

2)   (-∞ , 0]

3)   (-∞ , -6]

4)   [-5 , +∞)

More References and links

Find domain and range of functions,
Find the range of functions,
find the domain of a function and mathematics tutorials and problems.





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