Functions and Their Properties

⚡ Key Takeaways

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).

💡 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."

DefinitionFunction

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:

1

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

Derivation hint: collect every output produced as $x$ runs over the whole domain; it need not fill all of $B$ .

DefinitionInjective, Surjective, Bijective

Let $f : A \to B$ .

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

2

\quad f(x_1) = f(x_2) \;\Longrightarrow\; x_1 = x_2

Derivation hint: 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:

3

\quad \forall\, y \in B,\ \exists\, x \in A \ \text{such that}\ f(x) = y

Derivation hint: surjectivity is exactly the statement $\operatorname{ran}(f) = B$ .

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

DefinitionComposition and Inverse

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

4

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

Derivation hint: 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

5

\quad f^{-1}\big(f(x)\big) = x \quad\text{and}\quad f\big(f^{-1}(y)\big) = y

Derivation hint: the inverse undoes $f$ ; existence requires each output to come from exactly one input (bijectivity).

TheoremInvertibility Criterion

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

Proof

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. $\blacksquare$

∎

mathematician · 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.

🧠 Intuition

Think of three complementary pictures: 1. 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. 2. 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. 3. 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.

Figure

Vertical and horizontal line tests on the parabola

y = x^2

ExampleFinding domain, range, and inverse

Given: $f(x) = \sqrt{x - 2} + 3$ .

Step 1 — Domain. The radicand must be non-negative: $x - 2 \ge 0$ , so $x \ge 2$ . Domain $= [2, \infty)$ .

Step 2 — Range. Since $\sqrt{x-2} \ge 0$ , 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 $y = \sqrt{x-2} + 3$ . Then $y - 3 = \sqrt{x-2}$ , so $(y-3)^2 = x - 2$ , giving $x = (y-3)^2 + 2$ . Swap symbols:

6

\quad f^{-1}(x) = (x-3)^2 + 2, \qquad x \ge 3

Derivation hint: isolate the radical, square both sides, solve for $x$ ; restrict the new domain to the original range.

Step 4 — Check. $f(f^{-1}(3)) = f(2) = 3$ . ✓

\boxed{\text{Domain }=[2,\infty),\quad \text{Range }=[3,\infty),\quad f^{-1}(x)=(x-3)^2+2,\ x\ge 3}

ExampleComposition with numerical evaluation

Given: $f(x) = 2x + 1$ 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. $11 \ne 45$ , confirming $f \circ g \ne g \circ f$ in general.

\boxed{(f\circ g)(3) = 11, \qquad (g\circ f)(3) = 45}

⚠️ Common mistake

" $f^{-1}(x)$ means $1/f(x)$ ." No. The superscript $-1$ denotes the inverse function, not a reciprocal. The reciprocal is written $\big(f(x)\big)^{-1} = \dfrac{1}{f(x)}$ . These are entirely different objects: for $f(x)=2x$ , the inverse is $f^{-1}(x)=x/2$ , whereas the reciprocal is $1/(2x)$ .

⚠️ Common mistake

"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 $\mathbb{R}$ , but does on $[0,\infty)$ , namely $\sqrt{x}$ .

Counterexample

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 $y = +\sqrt{1-x^2}$ and $y = -\sqrt{1-x^2}$ produces two separate functions.

connection

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.

🔬 Where this appears

⏳ History

- 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.

open-problem

The naive notion "function = formula" collapses under scrutiny, raising deep questions still studied today. Which functions can be written explicitly? Most functions $\mathbb{R}\to\mathbb{R}$ 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.

Formal Statement

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also