Functions and Their Properties

A function is a rule that assigns to each input exactly one output. Key result: functions can be classified (injective, surjective, bijective), combined (sum, product, composition), inverted, and analyzed through properties like domain, range, symmetry, and monotonicity. Applies to every quantitative field — physics, economics, computer science, biology. Origins trace to Leibniz (1673) and were made rigorous by Dirichlet (1837).

A function $f$ from a set $A$ (the domain) to a set $B$ (the codomain) is a relation that pairs every element of $A$ with precisely one element of $B$ . This single-valued requirement is what distinguishes functions from arbitrary relations. Functions matter because they encode dependence: how one quantity is determined by another — distance by time, cost by quantity, temperature by altitude. A surprising fact: there exist functions that are continuous everywhere yet differentiable nowhere (the Weierstrass function), shattering the naive intuition that a "smooth-looking" curve must have a tangent at every point.

The Core Insight

A function is a deterministic machine: feed it a valid input and it returns one and only one output. Everything else — graphs, formulas, tables, inverses, transformations — is a different way of describing or manipulating that one constraint. The "vertical line test" is literally the geometric statement of "one output per input."

Formal Statement

Function

A function $f : A \to B$ is a rule assigning to each $x \in A$ exactly one element $f (x) \in B$ .

$A$ — the domain: the set of all permissible inputs.
$B$ — the codomain: the set in which outputs are declared to live.
$f (x)$ — the image of $x$ (the output corresponding to input $x$ ).
The range (or image of $f$ ) is the set of values actually attained:

ran (f) = {f (x) : x \in A} \subseteq B

⟳collect every output produced as

x

runs over the whole domain; it need not fill all of

B

Injective, Surjective, Bijective

Let $f : A \to B$ .

$f$ is injective (one-to-one) if distinct inputs give distinct outputs:

f (x_{1}) = f (x_{2}) ⟹ x_{1} = x_{2}

⟳negate "two different inputs share an output"; the contrapositive form is easiest to check.

$f$ is surjective (onto) if every element of the codomain is hit:

\forall y \in B, \exists x \in A such that f (x) = y

⟳surjectivity is exactly the statement

ran (f) = B

$f$ is bijective if it is both injective and surjective — a perfect pairing of $A$ with $B$ .

Composition and Inverse

The composition of $g : B \to C$ with $f : A \to B$ is

(g \circ f) (x) = g (f (x))

⟳apply

f

first, then feed its output into

g

; read right-to-left.

If $f$ is bijective, its inverse $f^{- 1} : B \to A$ satisfies

f^{- 1} (f (x)) = x and f (f^{- 1} (y)) = y

⟳the inverse undoes

f

; existence requires each output to come from exactly one input (bijectivity).

Invertibility Criterion

A function $f : A \to B$ has a two-sided inverse $f^{- 1} : B \to A$ if and only if $f$ is bijective.

We prove both directions.

Step 1 (Bijective $\Rightarrow$ invertible). Suppose

f

is bijective. For each

y \in B

, surjectivity (eq. [3]) guarantees at least one

x

with

f (x) = y

, and injectivity (eq. [2]) guarantees at most one such

x

. Hence exactly one

x

exists; define

f^{- 1} (y) = x

. This is a well-defined function and satisfies eq. [5] by construction. Step 2 (Invertible $\Rightarrow$ injective). Suppose

g = f^{- 1}

exists with

g (f (x)) = x

. If

f (x_{1}) = f (x_{2})

, apply

g

to both sides:

g (f (x_{1})) = g (f (x_{2}))

, so

x_{1} = x_{2}

. Thus

f

is injective. Step 3 (Invertible $\Rightarrow$ surjective). Take any

y \in B

. Let

x = g (y)

. Then

f (x) = f (g (y)) = y

by eq. [5]. So

y

is attained, proving surjectivity. Step 4. Steps 2–3 give bijectivity. Combined with Step 1, the equivalence holds.

■

Peter Gustav Lejeune Dirichlet | 1805–1859

Dirichlet gave the modern, formula-free definition of a function in 1837: $y$ is a function of $x$ if to each $x$ there corresponds a single definite $y$ , regardless of whether any algebraic expression connects them. His pathological example — the function equal to $1$ on rationals and $0$ on irrationals — demonstrated that "function" need not mean "nice formula," liberating analysis from the tyranny of explicit equations.

Building Intuition

Think of three complementary pictures:

Machine view — a box with an input slot and output slot. The rule "square it" turns $3 \mapsto 9$ and $- 3 \mapsto 9$ . Two inputs can share an output (that's fine), but one input can never produce two outputs.

Graph view — plot all points $(x, f (x))$ . The single-output rule becomes the vertical line test: any vertical line meets the graph at most once. Injectivity becomes the horizontal line test: any horizontal line meets the graph at most once.

Mapping view — draw two blobs (domain, codomain) and arrows from points to points. Each domain point has exactly one arrow leaving it. Injective = no two arrows land together; surjective = no codomain point is arrow-less.

Vertical and horizontal line tests on the parabola $y = x^2$

Worked Examples

Finding domain, range, and inverse

Given:

f (x) = x - 2 + 3

. Step 1 — Domain. The radicand must be non-negative:

x - 2 \geq 0

, so

x \geq 2

. Domain

= [2, \infty)

. Step 2 — Range. Since

x - 2 \geq 0

, the smallest output is at

x = 2

f (2) = 0 + 3 = 3

. As

x

grows,

f (x) \to \infty

. Range

= [3, \infty)

. Step 3 — Inverse. Set

y = x - 2 + 3

. Then

y - 3 = x - 2

, so

(y - 3)^{2} = x - 2

, giving

x = (y - 3)^{2} + 2

. Swap symbols:

f^{- 1} (x) = (x - 3)^{2} + 2, x \geq 3

⟳isolate the radical, square both sides, solve for

x

; restrict the new domain to the original range.

Step 4 — Check.

f (f^{- 1} (3)) = f (2) = 3

. ✓

Domain = [2, \infty), Range = [3, \infty), f^{- 1} (x) = (x - 3)^{2} + 2, x \geq 3

Composition with numerical evaluation

Given:

f (x) = 2 x + 1

and

g (x) = x^{2} - 4

. Find

(f \circ g) (3)

and

(g \circ f) (3)

. Step 1 — Compute $(f \circ g) (3)$ . Inner first:

g (3) = 3^{2} - 4 = 5

. Then

f (5) = 2 (5) + 1 = 11

. Step 2 — Compute $(g \circ f) (3)$ . Inner first:

f (3) = 2 (3) + 1 = 7

. Then

g (7) = 7^{2} - 4 = 45

. Step 3 — Observe non-commutativity.

11 \neq = 45

, confirming

f \circ g \neq = g \circ f

in general.

(f \circ g) (3) = 11, (g \circ f) (3) = 45

Common Misconceptions

" $f^{- 1} (x)$ means $1/ f (x)$ ." No. The superscript

- 1

denotes the inverse function, not a reciprocal. The reciprocal is written

(f (x))^{- 1} = \frac{1}{f ( x )}

. These are entirely different objects: for

f (x) = 2 x

, the inverse is

f^{- 1} (x) = x /2

, whereas the reciprocal is

1/ (2 x)

. "Every function has an inverse." Only bijective functions do (Theorem above). For non-injective functions you must restrict the domain before inverting — e.g.

f (x) = x^{2}

has no inverse on

R

, but does on

[0, \infty)

, namely

x

Claim: "If a graph is a curve, it represents a function." False. The circle $x^{2} + y^{2} = 1$ is a perfectly good curve, but the vertical line $x = 0$ crosses it at $(0, 1)$ and $(0, - 1)$ — two outputs for one input. The circle is a relation, not a function. Splitting it into $y = + 1 - x^{2}$ and $y = - 1 - x^{2}$ produces two separate functions.

Connections

Functions are the connective tissue of mathematics. They generalize the Linear Equations you solve in algebra into systematic input–output rules. The behavior of a function near a point is studied by Limits and Continuity, whose machinery underlies the Derivative (rate of change of a function) and the Integral (accumulation of a function). Special families — Polynomial Functions, Exponential and Logarithmic Functions, and Trigonometric Functions — each have signature properties. The idea of pairing sets bijectively powers the theory of Cardinality and Countability, where bijections measure the "size" of infinite sets.

Real-World Applications

Physics — projectile motion. Height as a function of time, $h (t) = h_{0} + v_{0} t - \frac{1}{2} g t^{2}$ , predicts when a ball lands (solve $h (t) = 0$ ).
Economics — supply and demand. Quantity demanded $Q_{d} (p)$ and supplied $Q_{s} (p)$ are functions of price; the equilibrium is where they intersect.
Computer science — hash functions. Map data of arbitrary size to fixed-size codes; ideal hashing approximates injectivity to avoid collisions.
Cryptography — RSA. Encryption uses a bijective modular-exponentiation function whose inverse is computationally hard without a secret key.
Medicine — pharmacokinetics. Drug concentration $C (t) = C_{0} e^{- k t}$ models exponential decay in the bloodstream, setting dosing schedules.
Engineering — transfer functions. A system's output is a function of its input frequency, central to control and signal processing.
Cartography — map projections. Bijective maps from the sphere to the plane (impossible without distortion) underlie every world map.

Historical Development

1673 — Gottfried Leibniz coins the word function (Latin functio) for quantities depending on a curve, such as slope and tangent length.
1718 — Johann Bernoulli describes a function as "a quantity composed in any manner of a variable and constants."
1748 — Leonhard Euler, in Introductio in analysin infinitorum, defines a function as an "analytic expression" and introduces the notation $f (x)$ .
1822 — Joseph Fourier shows arbitrary functions can be represented by trigonometric series, straining the "single formula" notion.
1837 — Dirichlet gives the modern set-correspondence definition, freeing functions from formulas.
1872 — Karl Weierstrass publishes a function continuous everywhere but differentiable nowhere, demolishing geometric intuition.
1895–1900s — Georg Cantor and others recast functions within set theory as sets of ordered pairs.
1914 — Felix Hausdorff formalizes the ordered-pair definition of a function as a special relation.
1930s — The Bourbaki group cements the modern domain–codomain–graph formalism used in textbooks today.

Subtleties & Limits

The naive notion "function = formula" collapses under scrutiny, raising deep questions still studied today. Which functions can be written explicitly? Most functions $R \to R$ have no finite description at all — there are uncountably many functions but only countably many finite formulas, so "almost every" function is indescribable. How wild can a function be? Weierstrass's nowhere-differentiable example and the existence (under the Axiom of Choice) of non-measurable functions show the concept stretches far beyond anything visualizable. Computability: Which functions can a machine actually evaluate? Alan Turing's work revealed functions (like the halting function) that are perfectly well-defined yet provably uncomputable — a boundary that remains central to theoretical computer science.

Functions and Their Properties

The Core Insight

Formal Statement

Function

Injective, Surjective, Bijective

Composition and Inverse

Invertibility Criterion

Peter Gustav Lejeune Dirichlet | 1805–1859

Building Intuition

Vertical and horizontal line tests on the parabola $y = x^2$

Worked Examples

Finding domain, range, and inverse

Composition with numerical evaluation

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also

Before this

Where this leads