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. In general, an absolute value function of the form \( y = |ax + b| \) has a range given by:
\( |ax + b| \geq 0 \)
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:
\[ f(x) = -|x| \]View Solution
Start with the range of the basic absolute value function:
\[ |x| \geq 0 \]Multiply both sides of the inequality by -1 and reverse the symbol of inequality to obtain:
\[ -|x| \leq 0 \]Hence, the range of \( -|x| \) is given by the interval:
\[ (-\infty, 0] \]Example 2
Find the range of function \( f \) defined by:
\[ f(x) = 2|2x + 4| \]View Solution
The range of the base absolute value expression is:
\[ |2x + 4| \geq 0 \]Multiply both sides of the inequality by 2 to write:
\[ 2|2x + 4| \geq 0 \]The range of the given function \( f \) can also be written in interval form as follows:
\[ [0, +\infty) \]Example 3
Find the range of function \( f \) defined by:
\[ f(x) = 2|-4x + 5| - 4 \]View Solution
The range of the base absolute value expression is:
\[ |-4x + 5| \geq 0 \]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 \( 2|-4x + 5| - 4 \) in interval form is:
\[ [-4, +\infty) \]Example 4
Find the range of function \( f \) defined by:
\[ f(x) = -\frac{1}{4}|-4x + 5| + \frac{1}{2} \]View Solution
The range of the base absolute value expression is:
\[ |-4x + 5| \geq 0 \]Multiply all terms of the inequality by \(-\frac{1}{4}\) and reverse the symbol of inequality to obtain:
\[ -\frac{1}{4}|-4x + 5| \leq 0 \]Add \(\frac{1}{2}\) to both sides of the above inequality to obtain:
\[ -\frac{1}{4}|-4x + 5| + \frac{1}{2} \leq \frac{1}{2} \]The range of values can be written in interval form as follows:
\[ (-\infty, \frac{1}{2}] \]