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Functions

Functions map a set of inputs to a set of outputs.

Each input maps to exactly one output, however an output value may have multiple possible inputs.

Terms

Term Description
domain The set of inputs
co-domain The set of outputs
range A subset of the co-domain with mapped values only
image A specific value in the co-domain that is the output for value in the domain
pre-image The value or values in the domain that map to a specific output

Notation

The expression \[ f: A \to B \] declares a function \(f\) that maps from the domain \(A\) to the domain \(B\).

When applied as \[ f: x \mapsto f(x) = y \] \(x \in X\) is the pre-image and \(y \in Y\) is the image.

In the case of \[ \begin{align} g: \mathbb R &\to \mathbb R \\ x &\mapsto x^2 + 1 \end{align} \] The function \(g\) accepts a real number as input, provides a real number as output and covers the range \([1, +\infty)\).

Common structures

The following are common structures for \(f : \mathbb R \to \mathbb R\).

Name Structure Conditions
linear \(f(x) = ax + b\)
quadratic \(f(x) = ax^2 + bx + c\) \(a \neq 0\)
exponential \(f(x) = b^x\) \(b > 0, b \neq 1\)

More on Expontential functions.

Properties

Injection

A function \(f:X \to Y\) is injective if and only if every element of the codomain \(Y\) is the image of at most one element of the domain \(X\).

Symbolicly: \[ f:X \rightarrowtail Y \iff \forall a,b \in X,\; f(a) = f(b) \implies a = b \] And conversely: \[ f:X \rightarrowtail Y \iff \forall a,b \in X,\; a \neq b \implies f(a) \neq f(b) \]

If an output is reachable from more than one input then a function is not injective.

For example, given \(g: \mathbb R \to \mathbb R\) then \(g(x) = x^2\) is not injective as \(g(5) = (5)^2 = (-5)^2 = g(-5)\). If however this is contrained as \(g: \mathbb R^+ \to \mathbb R\) then this function becomes injective.

Surjection

Surjective functions are functions with a range that is equivalent to the co-domain.

\[ f:X \twoheadrightarrow Y \iff \forall y \in Y,\; \exists x \in X,\; y = f(x) \] Alternatively expressed: every element of the co-domain of \(f\), \(Y\), has at least one pre-image in the domain of \(f\), \(X\).

Bijection

A function is bijective if it is both injective and surjective.

Symbolically this is expressed as: \[ f: X \leftrightarrow Y \iff \forall y \in Y,\; \exists! x \in X,\; y = f(x) \] Where \(\exists!\) means “there exists only one”.