Definition, domain and range, basic properties
Absolute value function
The absolute value function is the piecewise function:
f\left(x\right)=|x|=\begin{cases} x & \text{if }x\geq0 \cr \cr -x & \text{if }x\leq 0 \end{cases}
The absolute value function is the identity function for nonnegative values of x and ignores the negative sign for negative values of x.
The absolute value function evaluated at 2 is:
|2|=2
The absolute value function evaluated at -2 is:
|-2|=-\left(-2\right)=2
The domain of the absolute value function is the set of all real numbers \left(-\infty, \infty\right) whereas the range of the absolute value function is the set of all nonnegative numbers \left[0, \infty\right).
Graphical representation of the absolute value function
The absolute value function f\left(x\right)=|x| agrees:
- With the identity function y\longmapsto x when x\geq 0
- With the negative of the identity function y\longmapsto -x when x\leq 0
Hence, the graphical representation of the absolute function is as follows:
Equations and inequalities involving the absolute value function
Equations involving the absolute value function
If a is a nonnegative real number then the solutions of the equation |x|=a are a and -a.
Consider the equation:
|x|=3
The solutions are x=3 and x=-3.
This type of equation can also be solved graphically.
Consider again the equation:
|x|=3
To find solutions of the equation:
- Graph the function y \longmapsto |x| along with the horizontal line defined by the equation y=3.
- Find the points of intersection.
The x -values of the points of intersection correspond to solutions of the equation.
The solutions are x=3 and x=-3.
If a is a negative number then there are no solutions of the equation |x|=a.
Consider the equation:
|x|=-3
-3 is a negative number therefore the equation has no solutions.
This can also be observed by noticing the graph of y \longmapsto |x| and the horizontal line defined by the equation y=-3 have no intersection points.
The above rules for solving an equation involving the absolute value function provide a method to solve similar, but slightly more complicated equations.
- If k is a negative real number, \left| f\left(x\right) \right|=k has no solutions.
- If k is a nonnegative real number, \left| f\left(x\right) \right|=k if and only if f\left(x\right)=k or f\left(x\right)=-k.
Consider the equation:
|17x^5+3x^2-1|=-1
-1 is a negative number therefore the equation has no solution.
Consider the following equation:
|2x+3|=7
The equation is true if and only if one of the two following equations is true:
- 2x+3=7
- 2x+3=-7
The first equation is solved as follows:
2x+3=7\\2x=4\\x=2
The second equation is solved as follows:
2x+3=-7\\2x=-10\\x=-5
Therefore the equation |2x+3|=7 has two solutions: x=2 and x=5.
Inequalities involving the absolute value function
If a is nonnegative real number then |x|\geq a if and only if:
x\geq a or x\leq -a
The inequality |x|\geq 2 is true if and only if:
x\geq2 or x\leq-2
Therefore the inequality |x|\geq 2 has the solution set (-\infty, -2]\cup [2,\infty).
This type of inequality can also be solved graphically.
Considering the inequality |x|\geq 2:
Graph the function y\longmapsto |x| and then shade everything above the horizontal line defined by the equation y=2. The portion of the graph which lies in the shaded region corresponds to the solutions of the inequality.
Therefore the inequality |x|\geq 2 has solution set (-\infty, -2]\cup [2,\infty).
Similarly, if a is nonnegative real number then |x|>a if and only if:
x\gt a or x\lt -a
Consider the following inequality:
|x| \gt 2
The solution set of the inequality is the same as the solution set of the inequality |x|\geq 2 without the endpoints 2 and -2. Therefore the solution set is:
\left(-\infty, -2\right)\cup \left(2,\infty\right)
The above rules for solving inequalities involving absolute values allow us to solve more complex inequalities.
If f is a function and a nonnegative real number then the solution set of the inequality |f\left(x\right)| \gt a consists of x -values such that either f\left(x\right) \gt a or f\left(x\right) \lt -a.
Consider the following inequality:
|2x-3| \gt 3
x is a solution if and only if:
- 2x-3 \gt 3 or
- 2x-3 \lt -3
The first inequality is solved as follows:
2x-3 \gt 3\\2x \gt 6\\x \gt 3
The second inequality is solved as follows:
2x-3 \lt -3\\2x \lt 0\\x \lt 0
Therefore the solution set of the original inequality is
\left(-\infty, 0\right)\cup \left(3,\infty\right)
If a is a nonnegative real number then the inequality |x|\leq a has a solution set of the real numbers x such that:
-a\leq x\leq a
|x|\leq 3 if and only if:
-3\leq x\leq 3
The solution set of the inequality |x|\leq 3 is [-3{,}3].
The inequality |x|\leq 3 can be solved graphically. Graph the function y\longmapsto |x| and shade everything below the horizontal line y=3. The x -values on the graph in the shaded region correspond to the solution set of the inequality.
The solution set of the inequality |x|\leq 3 is [-3{,}3].
Similarly, if a is a nonnegative number, the inequality |x|\lt a has the solution set of the real numbers x such that:
-a \lt x\lt a
Consider the following inequality:
|2x-7| \lt 20
The inequality is solved as follows:
|2x-7| \lt 20\\-20 \lt 2x-7 \lt 20\\-13 \lt 2x \lt 27\\\dfrac{-13}{2} \lt x \lt \dfrac{27}{2}
The solution set is \left( \dfrac{-13}{2}, \dfrac{27}{2}\right)