# 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.

 Graphical Analysis of Range of Sine 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) 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 Find domain and range of functions, Find the range of functions, find the domain of a function and mathematics tutorials and problems.