Absolute Value Function

The absolute value function is defined by a rule which is divided into two cases.

The absolute value of x gives the distance between x and 0. It is always positive or zero. For example,

|3| = 3, |-3| = 3, |0|=0.

The domain of the absolute value function is the whole real line R while its range is the interval [0, ∞).

The absolute value function can also be described by the rule

Its graph is given by the equation y = . The graph is the V shown in Figure 1.2.10.

Figure 1.2.10

If a and b are two points on the real line, then from the definition of |x| we see that

Thus |a — b| is the difference between the larger and the smaller of the two numbers. In other words, |a - b| is the distance between the points a and b, as illustrated in Figure 1.2.11.

Figure 1.2.11

For example, |2 - 5| = 3, |4 - ( -4)| = 8. Here are some useful facts about absolute values.