1.2: Functions
- Page ID
- 22637
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)If \(A\) and \(B\) are sets, we call a relation \(R \subset A \times B\) a function with domain \(A\) if for every \(a \in A\) there exists one, and only one, \(b \in B\) such that \((a, b) \in R .\) We typically indicate such a relation with the notation \(f: A \rightarrow B,\) and write \(f(a)=b\) to indicate that \((a, b) \in R .\) We call the set of all \(b \in B\) such that \(f(a)=b\) for some \(a \in A\) the range of \(f .\) With this notation, we often refer to \(R\) as the graph of \(f\).
We say \(f: A \rightarrow B\) is one-to-one if for every \(b\) in the range of \(f\) there exists a unique \(a \in A\) such that \(f(a)=b .\) We say \(f\) is onto if for every \(b \in B\) there exists at least one \(a \in A\) such that \(f(a)=b .\) For example, the function \(f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}\) defined by \(f(z)=z^{2}\) is one-to-one, but not onto, whereas the function \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) defined by \(f(z)=z+1\) is both one-to-one and onto.
Given two functions, \(g: A \rightarrow B\) and \(f: B \rightarrow C,\) we define the composition, denoted \(f \circ g: A \rightarrow C,\) to be the function defined by \(f \circ g(a)=f(g(a))\).
If \(f: A \rightarrow B\) is both one-to-one and onto, then we may define a function \(f^{-1}: B \rightarrow A\) by requiring \(f^{-1}(b)=a\) if and only if \(f(a)=b\). Note that this implies that \(f \circ f^{-1}(b)=b\) for all \(b \in B\) and \(f^{-1} \circ f(a)=a\) for all \(a \in A .\) We call \(f^{-1}\) the inverse of \(f\).
Given any collection of nonempty sets, \(\left\{A_{\alpha}\right\}, \alpha \in I,\) we assume the existence of a function \(\phi: I \rightarrow B=\bigcup_{\alpha \in I} A_{\alpha},\) with the property that \(\phi(\alpha) \in A_{\alpha} .\) We call such a function a choice function. The assumption that choice functions always exist is known as the Axiom of Choice.