MA005: Calculus I

The absolute value function of a number $x$, $y = f(x) = | x |$, is the distance between the number $x$ and $0$. If $x$ is greater than or equal to $0$, then $| x |$ is simply $x – 0 = x$. If $x$ is negative, then $| x | is 0 – x = –x = –1 . x$ which is positive since $–1.$ (negative number) = a positive number. On some calculators and in some computer programming languages, the absolute value function is represented by $ABS(x)$.

Definition of $| x |$ : $| x | = \left\{ \begin{array}{ll} x & \mbox { if x ≥ 0 }\\ –x & \mbox { if x < 0 }\end{array} \right.$ or $| x | = \sqrt{x^2}$.

The domain of $y = f(x) = | x |$ consists of all real numbers. The range of $f(x) = | x |$ consists of all numbers larger than or equal to zero, all non–negative numbers. The graph of $y = f(x) = | x |$ (Fig. 10) has no holes or breaks, but it does have a sharp corner at $x = 0$. The absolute value will be useful later for describing phenomena such as reflected light and bouncing balls which change direction abruptly or whose graphs have corners.

The absolute value function has a number of properties which we will use later.

Properties of $| |$: For all real numbers $a$ and $b$:
(a) $| a | ≥ 0$. $| a | = 0$ if and only if $a = 0$.
(b) $| ab | = | a | | b |$
(c) $| a + b | ≤ | a | + | b |$

Taking the absolute value of a function has an interesting effect on the graph of the function. Since

$| x | = \left\{ \begin{array}{ll} x & \mbox { if x ≥ 0 }\\ –x & \mbox { if x < 0}\end{array} \right.$, then for any function $f(x)$ we have $| f(x) | = \left\{ \begin{array}{ll}f(x) & \mbox { if f(x) ≥ 0 }\\–f(x) & \mbox { if f(x) < 0 }\end{array} \right.$

In other words, if $f(x) ≥ 0$, then $| f(x) | = f(x)$ so the graph of $| f(x) |$ is the same as the graph of $f(x). If f(x) < 0$, then $| f(x) | = –f(x)$ so the graph of $| f(x) |$ is just the graph of $f(x)$ "flipped" about the x–axis, and it lies above the x–axis. The graph of $| f(x) |$ will always be on or above the x–axis.

Example 5: Fig. 11 shows the graph of $f(x)$. Graph (a) $| f(x) |$ , (b) $| 1 + f(x) |$ and (c) $1 + | f(x) |$.

Solution: The graphs are given in Fig. 12. In (b) we shift the graph of $f$ up 1 unit before taking the absolute value. In (c) we take the absolute value before shifting the graph up 1 unit.

Practice 7: Fig. 13 shows the graph of $g(x)$. Graph (a) $| g(x) |$ , (b) $| g(x – 1) |$ , and (c) $g( | x | )$.