The absolute value function

The absolute value function is a function defined by parts: $| x | = {\begin{array}{rcl} x & if & x \geq 0 \\ - x & if & x < 0 \end{array}$ Its domain $D o m (f) = R$ and its image $I m (f) = [0, + \infty)$ .

Example

Let's consider the function $f (x) = | 2 x - 1 |$ .

To represent it graphically, first, we will have to express it as a function defined by parts: $f (x) = | 2 x - 1 | = {\begin{array}{rcl} 2 x - 1 & if & 2 x - 1 \geq 0 \\ - (2 x - 1) & if & 2 x - 1 < 0 \end{array} = {\begin{array}{rcl} 2 x - 1 & if & x \geq \frac{1}{2} \\ 1 - 2 x & if & x < \frac{1}{2} \end{array}$ Now we can already represent it graphically considering, for example, points $0, \frac{1}{2}$ and $1$ :

$x$	$f (x)$
$0$	$1$
$\frac{1}{2}$	$0$
$1$	$1$

Therefore the function is (let's remember that only we need two points to represent a straight line):

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