Mathematics · Ch 2 — Relations and Functions

Standard Functions II — Modulus, Signum and Greatest Integer Functions, with Graphs

2.7

Standard Functions II — Modulus, Signum and Greatest Integer Functions, with Graphs

Modulus (absolute value) function. $f : \mathbb{R} \to \mathbb{R}$ defined by

f(x) = |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}

gives the numerical (non-negative) value of $x$ , discarding its sign. Domain $= \mathbb{R}$ ; range $= [0, \infty)$ , since $|x|$ is never negative but can be made as large as desired. Key properties used constantly in later chapters: $|x| \ge 0$ always; $|x| = |-x|$ ; $|xy| = |x||y|$ ; and the two-part definition above is what makes the modulus function piecewise rather than given by one single formula.

Signum function. $f : \mathbb{R} \to \mathbb{R}$ defined by

f(x) = \begin{cases} \dfrac{|x|}{x} = 1, & x > 0 \\ 0, & x = 0 \\ \dfrac{|x|}{x} = -1, & x < 0 \end{cases}

records only the sign of $x$ (positive, zero, or negative) as the single number $1$ , $0$ or $-1$ respectively, discarding the magnitude entirely. Domain $= \mathbb{R}$ ; range $= \{-1, 0, 1\}$ , only three values, however large or small $x$ becomes. …

Figure 2.7.1Graph of the modulus function f(x) = |x|

What this figure shows. A single V-shaped curve made of two straight-line rays meeting at a sharp corner exactly at the origin (0,0). The right-hand ray starts at the origin and rises at a 45-degree angle through the first quadrant (through points such as (1,1), (2,2)). The left-hand ray also starts at the origin and rises at a 45-degree angle but through the second quadrant, as x becomes more negative (through points such as (-1,1), (-2,2)) — so both rays climb away from the origin and neither ray ever goes below the x-axis; the whole graph sits on or abo …

Figure 2.7.2Graph of the signum function

What this figure shows. A graph made of three separate horizontal pieces, none of them joined to each other. For every x less than 0, a horizontal ray sits at height y=-1, drawn with an open (unfilled) circle at its right-hand end exactly above the origin, showing that the point (0,-1) itself is NOT part of the graph. At exactly x=0, a single filled (solid) dot is marked at the origin (0,0), the one point where the function equals 0. For every x greater than 0, a horizontal ray sits at height y=1, drawn with an open (unfilled) circle at its left-hand end exactly above the origin, showing that (0,1) is NOT part of the graph either. The visible picture is a low horizontal ray with an open end, a single solid dot at the origin, and a hig …

Figure 2.7.3Graph of the greatest integer function f(x) = [x]

What this figure shows. A 'staircase' made of a sequence of short horizontal line segments, each exactly one unit wide, stepping upward from left to right; within each segment, from an integer n (inclusive) up to but not including n+1, the height stays constant at y=n. Each segment is drawn with a filled (solid) dot at its left-hand end (at x=n, where the value [x]=n is actually attained) and an open (unfilled) circle at its right-hand end (approaching x=n+1, where the value would jump up to n+1 but has not yet done so) — so the graph looks like a rising set of disconnected steps, never a smooth or continuous line, with a vertica …

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