Functions:Definition

A function (or "mapping") is a relationship between two sets $A$ and $B$ that maps each input $a\in A$ to exactly one output $b\in B$. A function $f$ that maps elements of the set $A$ to elements in the set $B$ is denoted as $f:A\to B$, where $A$ is the domain of $f$ and $B$ is the codomain. We can also think of a function $f:A\to B$ as a set of ordered pairs $(a,b)$, $a\in A$ and $b\in B$, such that each element $a$ is paired with exactly one element $b$. If a function $f:A\to B$ maps an input $a$ to an output $b$, we can write that $f(a)=b$.

Note: a function cannot map one input to more than one output, but it can map more than one input to the same output. For example, let $A=\{a_{1},a_{2}\}$ and $B=\{b_{1},b_{2}\}$, $a_{1}\neq a_{2}$ and $b_{1}\neq b_{2}$. If $f:A\to B$ is a relation such that $f(a_{1})=b_{1}$ and $f(a_{1})=b_{2}$, then $f$ is NOT a function. However, a relation $g:A\to B$ such that $g(a_{1})=b_{1}$ and $g(a_{2})=b_{1}$ IS a valid function.

Examples:

• Let $A=\{1,2,3,4,5\}$ and $B=\{-1,3,4\}$, and let $f:A\to B$ such that $f(1)=-1$, $f(2)=3$, $f(3)=4$, $f(4)=3$, and $f(5)=-1$. Since each element of the domain maps to exactly one element (that is, there is no $f(a)=b_{1}$ and $f(a)=b_{2}$ such that $b_{1}\neq b_{2}$), $f$ is a function.
• For sets $A$ and $B$ in the previous example, let $g:A\to B$ be a relation such that $g(1)=1$, $g(1)=3$, $g(2)=4$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(4) = 4 } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(5) = 4 } . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g } maps the input Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 } to two distinct outputs, this relation is NOT a valid function.
• Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: \N\to\N } such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(n) = n + 1 } . This is a function, since each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\in\N } maps to exactly one element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1\in\N } .
• Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g: \N\to\Z } such that |g(n)| = n. This is not a valid function, since for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\in\N } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(n) <math> can equal both <math> n } and $-n$, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\neq -n } for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\neq 0 } .

Resources

• Course Textbook, pages 129-140
• Definition of Functions, Mathematics LibreTexts