Mathematics
Function Basics
A function in mathematics is a rule that assigns each input exactly one output. It can be represented as an equation, a graph, or a table. The input is also known as the independent variable, while the output is the dependent variable. Functions are fundamental to understanding relationships and patterns in mathematics.
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9 Key excerpts on "Function Basics"
eBook - PDFDiscrete Mathematics
Proofs, Structures and Applications, Third Edition
- Rowan Garnier, John Taylor(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
Chapter 5 Functions 5.1 Definitions and Examples In this chapter we consider another of the central concepts of modern mathematics, that of a function or mapping. Although functions have been used in mathematics for several centuries, it is only comparatively recently that a rigorous and generally accepted definition of the concept has emerged. When historians come to write the history of mathematics in the second half of the twentieth century, the rise in importance of functions of various kinds will almost certainly be one of their major themes. Like many of the concepts which we deal with in this book, that of a function is both simple and very general. Instead of giving the definition immediately, we shall begin with a notion with which you may very well be familiar from your previous studies—that of a (real) variable. Traditionally labelled x , a variable is often associated with expressions such as x 2 + 4 x -7 , 1 / ( x + 1) 3 , sin x, log x, etc. Expressions like these are frequently denoted f ( x ) and called ‘a function of (the variable) x ’. In such cases there is generally the assumption (which is often only implicit) that the variable x refers to an ‘arbitrary’ real number, although it may be subject to some restrictions, such as it must be positive. For us this idea of a function is both too restrictive and somewhat incomplete, although it does point towards a simpler and more general definition. The essence of the examples above is that we can calculate (in principle, at least) the value of the expression for any (allowed) value of the variable x . More important than an expression itself is the fact that it provides a ‘rule’ for calculating its value given any value of x . Two 220 Definitions and Examples 221 different expressions f ( x ) and g ( x ) may give the same values for all real numbers x , and we would regard the two expressions as defining the same function.
eBook - PDFExploring Mathematics
An Engaging Introduction to Proof
- John Meier, Derek Smith(Authors)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
5 Functions In this chapter we explore the concept of a mathematical function. Undoubtedly, func- tions have been prevalent in your previous mathematical experiences. You may have spent a lot of time working with functions given by an explicit formula, such as f (x) = x 2 sin(2x). In this chapter we develop a broader understanding of functions, presenting a commonly used and intuitive (but incomplete) definition in Section 5.1, followed by a variety of related terminology and concepts, and then a rigorous definition in Section 5.5. 5.1. What is a Function? The notion of a function is broader than numerical functions that are given by explicit formulas. D EFINITION 5.1 (Initial Version). Let A and B be sets. A function from A to B is a rule that assigns to each a ∈ A a unique associated element b ∈ B. There are two standard ways to represent “f is a function from A to B.” One is to write f : A → B, and the other is A f → B . Both notations are supposed to indicate that the function f takes an input element a ∈ A and outputs an element b ∈ B, and we express this by writing f (a) = b. In the sections that follow, you will often read statements like “Let f : A → B.” This is a short but complete sentence stating that f is a function from A to B. For example, consider the function g defined in a piecewise fashion for real numbers x: g(x) = −x 2 if x ≤ 0, x 4 if x > 0. The formulas −x 2 and x 4 both give real numbers if x is a real number, so we could use the notation g : R → R or R g → R because both A and B equal R. As another example, you might have encountered parametric curves which trace out patterns in the plane. One of these gives a circle parametrized by γ (t) = (sin(t), cos(t)) , as illustrated in Figure 1. Here the input is a real number t ∈ R and the output is a point (x, y) ∈ R 2 , so we write γ : R → R 2 or R γ → R 2 . 111 112 5. Functions 1 Figure 1. The image of the function γ (t) = (sin(t), cos(t)) is a circle.
eBook - PDF- Valentin Deaconu, Donald C. Pfaff(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
You already met real functions of real variables in calculus, like f (x) = x 2 , g(x) = ln x or h(x) = tan x. The domain and the set of values for these functions are subsets of R, and a function is defined as a formula (or algorithm) which associates to each input in the domain a precise output. The domain of f is R and the set of values is [0, ∞). The domain of g is (0, ∞) and the set of values is R. The domain of h is R \ {(2k + 1) π 2 : k ∈ Z} and the set of values is R. We will need to work with more general functions among all kinds of sets, not just subsets of the real numbers. Even though a function is a particular case of a relation, we study functions first and define relations in the next chapter. After giving the precise defini- tion of a function using its graph, we introduce operations and give several examples of functions. A given function determines two new functions, called the direct image and the inverse image, where the inputs and the outputs are sets. Sometimes it is necessary to shrink or enlarge the domain of a function, giving rise to restrictions and extensions. We also discuss one-to-one and onto functions, composition, and inverse functions. We conclude with families of sets and the axiom of choice, necessary in many proofs. 4.1 Definition and examples of functions Here is the formal definition of a function. Definition 4.1. A function from a set X to a set Y is a subset f of the Cartesian product X × Y such that for all x ∈ X there is a unique y ∈ Y with 〈x, y〉 ∈ f . The set X is called the domain of f , denoted dom(f ), and the set {y ∈ Y : ∃ x ∈ X with 〈x, y〉 ∈ f } 51 52 A bridge to higher mathematics is called the range of f , denoted ran(f ). The set Y is the set where f takes values, also called the codomain of f . Note that the range ran(f ) may be a proper subset of the codomain Y . We write f : X → Y , and for each x ∈ X the unique element y ∈ Y such that 〈x, y〉 ∈ f is denoted f (x).
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
A function may also be described through its relationship to other functions, for example, as the inverse function or a solution of a differential equation. There are uncountably many different functions from the set of natural numbers to itself, most of which cannot be expressed with a formula or an algorithm. In a setting where they have numerical outputs, functions may be added and multiplied, yielding new functions. Collections of functions with certain properties, such as continuous functions and differentiable functions, usually required to be closed under certain operations, are called function spaces and are studied as objects in their own right, in such disciplines as real analysis and complex analysis. An important operation on functions, which distinguishes them from numbers, is the composition of functions. Overview Because functions are so widely used, many traditions have grown up around their use. The symbol for the input to a function is often called the independent variable or argument and is often represented by the letter x or, if the input is a particular time, by the letter t . The symbol for the output is called the dependent variable or value and is often represented by the letter y . The function itself is most often called f , and thus the notation y = f ( x ) indicates that a function named f has an input named x and an output named y . ________________________ WORLD TECHNOLOGIES ________________________ A function ƒ takes an input, x , and returns an output ƒ( x ). One metaphor describes the function as a machine or black box that converts the input into the output. The set of all permitted inputs to a given function is called the domain of the function. The set of all resulting outputs is called the image or range of the function. The image is often a subset of some larger set, called the codomain of a function.
eBook - PDF- Alberto D. Yazon(Author)
- 2019(Publication Date)
- Arcler Press(Publisher)
Fundamentals of Mathematics 1 CONTENTS 1.1. Introduction ........................................................................................ 2 1.2. Proof And Mathematical Argument ..................................................... 5 1.3. Sets, Relations, And Functions .......................................................... 10 1.4. Construction And Properties Of Number Systems ............................. 21 1.5. Some Number Theory ....................................................................... 27 1.6. Case Study: Analysis On Mathematics Fundamental Knowledge For Mathematics Engineering Courses Based on A Comparative Study Of Students’ Entry Performance ................ 30 References ............................................................................................... 35 Fundamentals of Advanced Mathematics 2 The learning of mathematics in schools represents a basic preparation for adult life and a gateway to a vast array of career choices. In a social perspective, proficiency in mathematics is essential for functioning in everyday life, as well as for success in increasingly technology-based workplace. Students who take higher-level mathematics and science courses which require strong fundamental skills in mathematics are more likely to attend and to complete college. The globalization of markets, the spread of information technologies, and the premium paid for workforce skills all emphasize the growing need for proficiency in mathematics. There is a widespread interest among industrialized countries in improving the levels of mathematics achievement in schools. Apart from the economic benefits, it is argued that this would bring about economic progress by better preparing young people for the numeracy demands of modern workplaces, and raising the overall skill levels of the workforce.
eBook - PDF- Scott A. Taylor(Author)
- 2023(Publication Date)
- American Mathematical Society(Publisher)
Chapter 8 Functions Key Terms • definition of a function and inverse function • domain, codomain, range, graph of a function • identity, constant, and coordinate functions • function composition • finite and infinite sequence • injective, surjective, bijective function • inverse function It is not necessary that be subject to the same rule as regards throughout the interval; indeed, one need not even be able to express the relationship through mathematical operations ...It doesn’t matter if one thinks of this [correspondence] so that different parts are given by different laws or desig- nates it [the correspondence] entirely lawlessly. —Lejeune Dirichlet 1 8.1. The definition of a function In searching out this matter, I found it by no means clearly laid down what is meant by the solution of a differential equation: and, on looking further, I found some degree of ambiguity attaching to the word “equation” itself. —Augustus De Morgan 2 1 Dirichlet (1805–1859) made important contributions to both number theory and analysis. Reprinted from The Gen- esis of Point Set Topology, Jerome H. Manheim, pp. 52–53, Copyright ©1964. Used with permission. 2 Quotation from On the question, What is the Solution of a Differential Equation? Trans. Camb. Phil. Soc (10) 1864. 181 182 8. Functions If and are sets, a function ∶ → is a way of transforming the elements of into elements of . Beginning calculus courses study functions ∶ ℝ → ℝ, what it means to graph such functions, and how to determine if they are continuous or differ- entiable. More advanced calculus classes study differentiable functions ∶ ℝ 2 → ℝ and ∶ ℝ 3 → ℝ (and perhaps other types of functions). Linear algebra classes study linear functions ∶ ℝ → ℝ . In this text, we study functions in complete general- ity. The key concepts from this section are used throughout theoretical and applied mathematics. In Chapter 6 we gave a formal definition of “function” in terms of sets.
eBook - PDF- Daniel Ashlock, Colin Lee(Authors)
- 2022(Publication Date)
- Springer(Publisher)
85 C H A P T E R 7 Functions While every reader of this text is likely familiar with functions such as a quadratic function like f .x/ D x 2 C 2x C 1, this is likely a student’s first real introduction to abstract mathematical functions (or set theoretic functions). In more advanced mathematics the set theoretic defini- tion of functions is used as the default definition of a function. In this section we will define functions and extend much of our ability to work with sets to infinite sets. There are a number of different types of functions and so this section contains a great deal of terminology. Recall that two finite sets are the same size if they contain the same number of elements. It is possible to make this idea formal by using functions and, once the notion is formally defined, it can be applied to infinite sets. 7.1 MATHEMATICAL FUNCTIONS Definition 7.1 An ordered pair is a collection of two elements with the added property that one element comes first and one element comes second. The set containing only x and y (for x ¤ y ) is written fx; y g. The ordered pair containing x and y with x first is written .x; y/. Notice that while fx; xg is not a well-defined set, .x; x/ is a well-defined ordered pair because the two copies of x are different by virtue of coming first and second. The reason for defining ordered pairs at this point is that it permits us to make an important formal definition that pervades the rest of mathematics. Definition 7.2 A function f with domain S and range T is a set of ordered pairs .s; t/ with first element from S and second element from T that has the property that every element of S appears exactly once in some ordered pair. We write f W S ! T for such a function. We say f is a mapping of S to T , or that f maps S to T . Example 7.3 Suppose that A D fa; b; c g and B D f0; 1g then f D f.a; 0/; .b; 1/; .c; 0/g
eBook - PDF- Cynthia Y. Young(Author)
- 2018(Publication Date)
- Wiley(Publisher)
We will discuss common functions, the domain and range of functions, and graphs of functions. We will determine whether a function is increasing, decreasing, or constant on an interval and calculate the average rate of change of a function. We will perform operations on functions and the composition of functions. Finally, we will discuss one-to-one functions and inverse functions. FUNCTIONS AND THEIR GRAPHS 1.1 FUNCTIONS 1.2 GRAPHS OF FUNCTIONS 1.3 GRAPHING TECHNIQUES: TRANSFORMATIONS 1.4 COMBINING FUNCTIONS 1.5 ONE-TO-ONE FUNCTIONS AND INVERSE FUNCTIONS • Definition of a Function • Functions Defined by Equations • Function Notation • Domain of a Function • Recognizing and Classifying Functions • Increasing and Decreasing Functions • Average Rate of Change • Piecewise- Defined Functions • Horizontal and Vertical Shifts • Reflection about the Axes • Stretching and Compressing • Adding, Subtracting, Multiplying, and Dividing Functions • Composition of Functions • One-to-One Functions • Inverse Functions • Graphical Interpretation of Inverse Functions • Finding the Inverse Function 102 CHAPTER 1 Functions and Their Graphs 1.1.1 Definition of a Function What do the following pairs have in common? ■ ■ Every person has a blood type. ■ ■ Temperature is some typical value at a particular time of day. ■ ■ Every working household phone in the United States has a 10-digit phone number. ■ ■ First-class postage rates correspond to the weight of a letter. ■ ■ Certain times of the day are start times for sporting events at a university. They all describe a particular correspondence between two groups. A relation is a correspondence between two sets. The first set is called the domain and the corresponding second set is called the range. Members of these sets are called elements. 1.1.1 SKILL Determine whether a relation is a function. 1.1.1 CONCEPTUAL Understand that all functions are relations, but not all relations are functions.
eBook - PDFDiscrete Mathematics
Mathematical Reasoning and Proof with Puzzles, Patterns, and Games
- Douglas E. Ensley, J. Winston Crawley(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
27. In your own words, explain how you can tell from a list of ordered pairs whether the relation with that list as its rule is a function. 28. Explain how you can tell from a list of ordered pairs whether the inverse of the relation with that list as its rule is a function. 4.2 The Composition Operation The idea of composition of functions and relations is a very basic one. Mathematics is largely about how complex concepts, structures, and properties can be built logically out of simpler ones. Since functions and relations are fundamental structures in mathematics, it stands to reason that combinations of two or more of these structures could be important. Composition of Functions Before being swept away by a formal definition, let us consider an example of how composition naturally arises in the English language. One rule that relates pairs of people is the “husband of” relation. Another rule of this type is the “mother of” relation. These relations can be combined to give two distinct meanings: • The relation “mother of the husband of” associates a woman with her mother-in- law. • The relation “husband of the mother of” associates any person with his or her father or stepfather. The preposition “of” naturally ties together English clauses in the same way that composition ties together mathematical functions, as we see in our formal definition. Definition If f : A → B and g : B → C , then we can build a new function called (g ◦ f ) that has domain A and codomain C , and that follows the rule (g ◦ f )(x) = g( f (x)). We call (g ◦ f ), read “g of f ,” the composition of g with f . 4.2 The Composition Operation 269 The double of the square root g f Description order for (g ° f ) g ( f ( x )) Second First Evaluation order for (g ° f ) Figure 4-22 How to read (g ◦ f ). Example 1 Given the function f : R ≥0 → R defined by the rule f (x) = √ x, and the function g : R → R defined by the rule g( y) = 2 · y, describe the domain, codomain, and rule for the function (g ◦ f ).
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