Functions and Mathematical Models

Calculus studies how one quantity changes with another, so its basic object is a rule that ties the two together. That rule is a function. Before differentiating or integrating anything, we need a precise account of what a function is, which functions occur often enough to name, and how to assemble complicated functions from simple parts.

Definition of a function

The one-output-per-input requirement is the whole content of the definition. A symbol standing for an arbitrary element of the domain is the independent variable; a symbol standing for a value in the range is the dependent variable. When a function is given by a formula with no stated domain, the convention is that the domain is the set of all inputs for which the formula produces a real number. For $f (x) = x + 2$ that forces $x + 2 \geq 0$ , so the domain is $[- 2, \infty)$ ; for $g (x) = 1/ (x^{2} - x)$ the excluded inputs are the roots of the denominator, giving domain $(- \infty, 0) \cup (0, 1) \cup (1, \infty)$ .

A function admits four representations, and switching among them is routine:

Verbal — a description in words (the cost of mailing an envelope of weight $w$ ).
Numerical — a table of input-output pairs, as from an experiment.
Visual — a graph, the set ${(x, f (x)) : x \in D}$ of points in the plane.
Algebraic — an explicit formula.

The graph is the most common picture, and it carries the domain and range directly: the domain is the shadow of the graph on the $x$ -axis, the range its shadow on the $y$ -axis. Reading a value off a graph amounts to measuring the height of the curve above a point $x$ .

Two vertical intersections at $x = a$ would be two outputs assigned to the same input, which the definition forbids. The parabola $x = y^{2} - 2$ fails the test; it splits into the two functions $y = x + 2$ and $y = - x + 2$ , its upper and lower halves.

A function need not be given by a single formula. A piecewise defined function uses different formulas on different parts of its domain. The absolute value is the standard case,

∣ x ∣ = {x - x if x \geq 0, if x < 0,

and step functions such as postage cost jump from one constant value to the next.

The difference quotient

One combination built from a single function is common enough to name. For $f$ and a nonzero increment $h$ , the difference quotient

\frac{f ( a + h ) - f ( a )}{h}

is the average rate of change of $f$ between $x = a$ and $x = a + h$ . For $f (x) = 2 x^{2} - 5 x + 1$ a direct expansion gives

\frac{f ( a + h ) - f ( a )}{h} = \frac{( 2 a ^{2} + 4 ah + 2 h ^{2} - 5 a - 5 h + 1 ) - ( 2 a ^{2} - 5 a + 1 )}{h} = 4 a + 2 h - 5.

Letting $h$ shrink toward $0$ turns this average rate into an instantaneous one; that passage to the limit defines the derivative.

A catalog of essential functions

A few function families model most elementary phenomena, and their graphs form a useful reference set.

Six elementary families on a common frame: a line, a parabola, a square root, a reciprocal, the sine, and a growing exponential.

The families divide by the algebra used to build them:

Linear: $f (x) = m x + b$ . The graph is a line; the defining feature is a constant rate of change $m$ , so equal steps in $x$ produce equal steps in $y$ .
Polynomial: $P (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}$ with $a_{n} \neq = 0$ of degree $n$ . Domain $(- \infty, \infty)$ . Degree $1$ is linear, degree $2$ quadratic (a parabola), degree $3$ cubic.
Power: $f (x) = x^{a}$ . Integer $a$ gives the monomials; $a = 1/ n$ gives root functions $n x$ ; $a = - 1$ gives the reciprocal $1/ x$ , whose graph is a hyperbola.
Rational: a ratio $P (x) / Q (x)$ of polynomials, defined wherever $Q (x) \neq = 0$ .
Trigonometric: $sin x$ , $cos x$ , $tan x$ , and their reciprocals; periodic, with $sin$ and $cos$ bounded in $[- 1, 1]$ .
Exponential and logarithmic: $f (x) = b^{x}$ with base $b > 0$ , and its inverse $log_{b} x$ . Exponentials model growth and decay and are examined in their own module.

Family	Form	Domain	Distinguishing trait
Linear	$m x + b$	$R$	constant rate of change
Polynomial	$\sum a_{k} x^{k}$	$R$	finitely many turning points
Power	$x^{a}$	depends on $a$	roots, reciprocals as special cases
Rational	$P / Q$	$Q \neq = 0$	vertical asymptotes at zeros of $Q$
Trigonometric	$sin x, cos x, \dots$	$R$ (some restricted)	periodic
Exponential	$b^{x}$	$R$	constant ratio per unit step

The distinction between polynomial and exponential growth is worth stating precisely. A linear function adds a fixed amount per unit step; an exponential multiplies by a fixed ratio per unit step. The multiplicative rule wins in the long run: an exponential eventually overtakes every polynomial, a fact made quantitative by l'Hospital's Rule.

Fitting a model to data

A mathematical model is a function chosen to describe a real phenomenon. Given a scatter of data points, one selects a family whose shape matches the trend and then fits its parameters. Points falling near a line call for a linear model; a parabolic arc calls for a quadratic. For carbon-dioxide concentration measured yearly, the points lie close to a line, and the least-squares fit $C \approx 1.71262 t - 3054.14$ (with $t$ the year) reproduces the record: it returns $348.8$ ppm for $1987$ , against a measured $348.9$ . Reading a model inside the observed range is interpolation and is reliable; reading it outside is extrapolation and is not, since nothing guarantees the trend continues.

New functions from old

Three operations generate the endless variety of functions in calculus from the catalog above: transforming one function, combining two arithmetically, and composing one inside another.

Transformations

Adding or multiplying constants shifts, stretches, or reflects a graph in predictable ways. Take a base function $y = f (x)$ and a constant $c > 0$ .

The parabola

y = x^{2}

shifted up by

c

, shifted right by

c

, and reflected across the

x

-axis; each constant acts on one coordinate.

The rules split cleanly by whether the constant touches the output or the input:

Operation	Effect on the graph
$f (x) + c$	shift up by $c$
$f (x) - c$	shift down by $c$
$f (x - c)$	shift right by $c$
$f (x + c)$	shift left by $c$
$c f (x)$ , $c > 1$	stretch vertically by $c$
$f (c x)$ , $c > 1$	compress horizontally by $c$
$- f (x)$	reflect across the $x$ -axis
$f (- x)$	reflect across the $y$ -axis

Constants applied to the output act as expected; constants applied to the input act in reverse, because $f (x - c)$ reaches a given height only once $x$ has advanced to $x - c =$ the point where $f$ had it. A factor $c$ inside the argument compresses the graph horizontally by $c$ : $y = sin 2 x$ runs through a full cycle in half the span of $y = sin x$ .

Horizontal compression:

y = sin 2 x

completes a cycle in half the run of

y = sin x

because the input advances twice as fast.

Completing the square rewrites any quadratic as a shifted parabola, exposing the transformations directly.

A related operation reflects part of a graph rather than the whole. Taking the absolute value of a function leaves the part above the $x$ -axis fixed and flips the part below it upward, since $∣ f (x) ∣ = f (x)$ where $f (x) \geq 0$ and $∣ f (x) ∣ = - f (x)$ where $f (x) < 0$ .

The dashed parabola

y = x^{2} - 1

; taking the absolute value reflects its below-axis arch upward, giving the solid graph of

y = ∣ x^{2} - 1∣

Combinations

Two functions $f$ and $g$ combine pointwise under the arithmetic operations, on the intersection of their domains (and away from zeros of the denominator for the quotient):

(f + g) (x) = f (x) + g (x), (f g) (x) = f (x) g (x), (\frac{f}{g}) (x) = \frac{f ( x )}{g ( x )} .

Composition

The operation with no counterpart in ordinary arithmetic feeds the output of one function into the input of another.

Read $f \circ g$ right to left: apply $g$ first, then $f$ . The order matters, since $f \circ g$ and $g \circ f$ are generally different functions.

Composition as two machines in series:

x

enters

g

, its output

g (x)

becomes the input to

f

, and the final output is

f (g (x))

For $f (x) = x$ and $g (x) = x^{2} + 1$ ,

(f \circ g) (x) = x^{2} + 1, (g \circ f) (x) = (x)^{2} + 1 = x + 1,

with domains $(- \infty, \infty)$ and $[0, \infty)$ respectively. Composition is the structure the Chain Rule differentiates.

The domain of a composite is not the whole real line by default; each stage imposes its own restriction, and both must hold.

Recognizing a function as a composite runs the machine backward: given a formula, name the stages applied in order.

Symmetry

Two symmetry conditions simplify graphing and, later, integration.

An even function reflects across the

y

-axis (left); an odd function carries to itself under a half-turn about the origin (right).

The power functions supply the archetypes and the naming: $f (x) = x^{n}$ is even exactly when $n$ is even and odd exactly when $n$ is odd. Most functions are neither; $h (x) = 2 x - x^{2}$ satisfies neither identity. Knowing a function is even or odd halves the work of graphing it, since one side determines the other.

Stewart, §1.1 — the rule-based definition of a function, its domain, range, and the four representations (verbal, numerical, visual, algebraic). ↩
Stewart, §1.3 — shifts, stretches, and reflections; completing the square to graph a quadratic; the absolute value of a function; and composition with its domain. ↩