Function Equations and definitions

Definition

A function $f:X\to Y$ from a set $X$ to a set $Y$ assigns to every $x\in X$ exactly one element $f(x)\in Y$.

• $X=\mathrm{dom}(f)$ is the Domain.
• $Y$ is the codomain; see Codomain and range.
• $f(X)=\{f(x):x\in X\}$ is the image/range.

Graph

The graph of a function is the set ${\Gamma }_f=\{(x,f(x)):x\in X\}\subseteq X\times Y.$ This is why functions can be treated as special relations.

Composition

If $f:X\to Y$ and $g:Y\to Z$, then $g\circ f:X\to Z,(g\circ f)(x)=g(f(x)).$ Composition order matters.

Mapping properties

• Injective: $f(x_1)=f(x_2)\Rightarrow x_1=x_2$.
• Surjective: for every $y\in Y$, there exists $x\in X$ such that $f(x)=y$.
• Bijective: injective and surjective.

Inverse

If $f$ is bijective, then there is an inverse $f^{-1}:Y\to X$ satisfying

\qquad f\circ f^{-1}=\operatorname{id}_Y.$$ See [[Inverse functions]] for examples and domain restrictions. ## Composition and identity Function composition means feeding one function into another:

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

$$Theidentityfunctiondoesnothing:$$

\operatorname{id}_A(x)=x.

A function $f:A o B$ has an inverse exactly when there is a function $f^{-1}:B o A$ such that

f^{-1}\circ f=\operatorname{id}_A,\qquad f\circ f^{-1}=\operatorname{id}_B.

$$Thislinksdirectlyto[[Inversefunctions]]and[[Bijectivefunction]].$$