9: An Overview of Functions
- Page ID
- 173415
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 9.1: Relations and Mappings
- This page covers ordered pairs and relations in mathematics, emphasizing the significance of order in pairs \((a,b)\) and the Cartesian product \(A \times B\), which derives pairs from two sets. It explains that the cardinality of the product is the product of individual set sizes and defines relations as subsets of these products, including discussions on domains and ranges.
- 9.2: The Definition of a Function
- This page introduces the concept of functions in mathematics, defining them as relations where each input has one output. It explains that a single input cannot have multiple outputs and presents the Vertical Line Test to assess if a graph represents a function. Through examples, the page illustrates how to identify functions from ordered pairs, equations, and tables, as well as demonstrates applying the Vertical Line Test to graphs.
- 9.3: Evaluating Functions
- This page covers function notation and evaluation in mathematics, defining key terms like \(y = f(x)\), where \(f\) is the function name, \(x\) is the independent variable, and \(f(x)\) is the dependent variable's value. It clarifies that \(f(x)\) does not denote multiplication and outlines how to evaluate functions using constants, fractions, and algebraic expressions.
- 9.4: The Domain and Range of a Function
- This page explains domain and range in mathematical functions, defining the domain as the set of all possible input values and the range as all possible outputs. It includes key concepts like the natural domain and restrictions such as division by zero and even-indexed radicals. Examples demonstrate how to determine the domain and range of different function types—polynomials, rational, and radical functions—using algebraic techniques and graphical analysis.
- 9.5: Function Arithmetic and Domain
- This page discusses the arithmetic of functions, including how to combine them through addition, subtraction, multiplication, and division, while highlighting the importance of understanding their domains and exclusions. It provides examples of combined functions and illustrates the simplification of fractional expressions, emphasizing the resulting domain restrictions due to specific values that must be excluded. This ensures a thorough grasp of domain implications even after simplification.
- 9.6: The Difference Quotient
- This page covers the average rate of change and difference quotient in functions, defining the former over an interval and the latter at a specific point. It includes theorems for linear and quadratic functions, emphasizes careful evaluation and simplification, and provides multiple examples. The content culminates in highlighting the relationship between the difference quotient and the average rate of change, offering a comprehensive understanding of these concepts.
- 9.7: Function Composition and Domain
- This page explains the composition of functions, describing it as using the output of one function as the input for another, represented as \(f \circ g\). It highlights that the domain of the composite function includes values of \(x\) for which both \(g(x)\) and \(f\) are defined.
- 9.8: Inverse Functions
- This page explains inverse functions, defining one-to-one functions and key theorems such as the Horizontal Line Test, which determine when a function has an inverse. It clarifies the notation \(f^{-1}\) is not a reciprocal and outlines methods for finding inverses along with domain/range relationships.
- 9.9: Restricting Domains and Forcing Invertibility
- This page covers the concept of making functions invertible through restriction to ensure they are one-to-one, highlighting that restricted functions retain their rules while focusing on a smaller domain. It provides examples demonstrating how to restrict common functions like fourth-degree polynomials, absolute value, and sine functions to maintain uniqueness, allowing for the calculation of their inverses.