Mathematics · Ch 2 — Relations and Functions

Standard Functions I — Constant, Identity, Polynomial and Rational Functions, with Graphs

2.6

Standard Functions I — Constant, Identity, Polynomial and Rational Functions, with Graphs

Constant function. $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = c$ for every $x \in \mathbb{R}$ , where $c$ is a fixed real number, is called a constant function. Domain $= \mathbb{R}$ ; range $= \{c\}$ , a single value. Its graph is the horizontal straight line $y = c$ , parallel to the $x$ -axis and at (signed) height $c$ above it.

Identity function. $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x$ for every $x \in \mathbb{R}$ is called the identity function — every element is sent to itself. Domain $= \mathbb{R}$ ; range $= \mathbb{R}$ . Its graph is the straight line $y = x$ , passing through the origin at exactly $45°$ to both axes, i.e. bisecting the first and third quadrants.

Polynomial function. $f(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$ , where $n$ is a non-negative integer and $a_0, a_1, \dots, a_n$ are real constants (with $a_n \ne 0$ if $n \ge 1$ ), is called a polynomial function of degree $n$ . Constant functions ( $n=0$ ) and the identity function (degree $1$ , $f(x)=x$ ) are themselves the simplest polynomial functions. Domain $= \mathbb{R}$ always, since every operation involved (multiplication, addition of real numbers) is defined for every real $x$ . The range depends on the degree and coefficients: a non-constant linear function $f(x) = ax+b$ ( $a \ne 0$ ) has range $\mathbb{R}$ ; a quadratic $f(x) = ax^2+bx+c$ has graph a parabola and range either $[k, \infty)$ or $(-\infty, k]$ for the $y$ -value $k$ at its vertex, according to the sign of $a$ . …

Figure 2.6.1Graph of the constant function f(x) = c

What this figure shows. A single horizontal straight line drawn at a fixed height c above (or below, if c is negative) the x-axis, running the full width of the visible plane and staying at exactly that height for every x-value; the line never rises or falls, is parallel to the x-axis, and would coincide with the x-axis itself only in the special case c = 0. A sample point is marked on the line at some x-value with its …

Figure 2.6.2Graph of the identity function f(x) = x

What this figure shows. A single straight line passing exactly through the origin (0,0) and rising at a 45-degree angle to the x-axis, so that it bisects the first and third quadrants; for every x-value the marked point on the line sits at height y equal to that same x-value (e.g. the points (1,1), (2,2), (-1,-1) are marked lying exactly on the line), with no curvature anywhere …

Figure 2.6.3Graph of a quadratic polynomial function, e.g. f(x) = x^2

What this figure shows. A single upward-opening, symmetric U-shaped curve (a parabola) with its lowest point (vertex) sitting exactly at the origin (0,0); the curve rises steadily and symmetrically on both sides of the y-axis as x moves away from 0 in either direction, is symmetric about the y-axis (the left and right halves are mirror images of each other), and never dips below the x-axis, touching it only at the single verte …

Figure 2.6.4Graph of a rational function, e.g. f(x) = 1/x

What this figure shows. Two separate curved branches, never touching each other or the axes. The right-hand branch lies entirely in the first quadrant (x>0, y>0): as x increases from just above 0 the curve falls steeply from very high up, flattening out and approaching the x-axis (but never touching it) as x grows large; as x approaches 0 from the right the curve rises steeply upward, approaching the y-axis (but never touching it) without limit. The left-hand branch lies entirely in the third quadrant (x<0, y<0) and is the point-symmetric mirror image of the right-hand branch through the origin, approaching the negative x- …

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