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How can the absolute value function be written using different rules for positive and negative inputs?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

The absolute value function, denoted as $| x |$ , measures the magnitude of a real number and represents its distance from zero on a number line. Because distance is always non-negative, the function is defined using a piecewise rule that accounts for the sign of the input.1

The absolute value function can be written as:

$| x | = {\begin{matrix} x & if x \geq 0 \\ - x & if x < 0 \end{matrix}$

In this definition, if the input $x$ is positive or zero, the output is simply $x$ . If the input is negative, the output is the negation of $x$ ( $- x$ ), which results in a positive value. This approach of defining a function using different rules for discrete intervals is known as a piecewise definition.12