# Functions

Let \(X\) and \(Y\) be sets and \(P(x,y)\) be a predicte depending on \(x \in X\) and \(y \in Y\) such that \(\forall x \in X\) there is exactly one \(y \in Y\) for which \(P(x,y)\) is true. A **function** \(f : X \rightarrow Y\) defined by \(P(x,y)\) maps any input \(x \in X\) to an output \(f(x) \in Y.\)

\(X\) is the **domain** and \(Y\) is the **codomain** or **range** of the function \(f.\)

Two functions with the same domain \(X\) and codomain \(Y,\) \(f : X \rightarrow Y\) and \(g : X \rightarrow Y,\) are **equal** \(f=g\) if and only if \(f(x)=g(x)\, \forall x \in X.\)

Let \(f : X \rightarrow Y\) and \(g : Y \rightarrow Z\) be two functions such that the codomain of \(f\) is the same set as the domain of \(g.\) The **composition** \(g \circ f : X \rightarrow Z\) of the two functions \(f\) and \(g\) if the output of \(g\) when its input is the output of \(f\) given an input \(x \in X.\)

If the codomain of \(f\) does not match the domain of \(g,\) the composition \(g \circ f\) is undefined.

A function \(f: X \rightarrow Y\) is **injective** or “one-to-one” if distinct elements of its domain \(X\) map to distinct elements of its codomain \(Y.\)

A function \(f: X \rightarrow Y\) is **surjective** or “onto” if all elements of its domain \(X\) map to all elements of its codomain \(Y.\)

A function \(f: X \rightarrow Y\) is **bijective** or **invertible** if it is both injective and surjective.

If \(f: X \rightarrow Y\) is a function with domain \(X\) and codomain \(Y\) and \(S \subseteq X,\) the **image** of \(S\) under \(f,\) \(f(S),\) is the subset of elements in \(Y\) that were mapped from \(S.\)

If \(S \subseteq Y,\) the **inverse image** of \(S\) under \(f,\) \(f^{-1}(S),\) is the subset of elements in \(X\) which map to \(S\) by \(f.\)