Functions
In mathematics, a function is a relation between a set of inputs and a set of possible outputs, where each input is related to exactly one output. Functions are often represented by equations or graphs and are used to model various real-world phenomena. They are fundamental to calculus, algebra, and other branches of mathematics, providing a way to describe and analyze relationships between quantities.
8 Key excerpts on "Functions"
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The slope of a parabola changes along the curve. Linear and quadratic Functions are special types of algebraic Functions. When an algebraic function takes the form of an equation, the equation is typically written in the explicit form y = … in which the right side of the equation is obtained by performing some combination of basic algebraic operations on x. These operations include addition, multiplication, division, raising to a power, and taking the n th root. LESSONS IN CHAPTER 4 • Lesson 4-1: Function Concepts • Lesson 4-2: Quadratic Functions and Their Graphs • Lesson 4-3: Solving Quadratic Equations • Lesson 4-4: Solving a Linear-Quadratic System • Lesson 4-5: Applying Quadratic Equations • Lesson 4-6: Solving Quadratic Inequalities Lesson 4-1: Function Concepts KEY IDEAS Functions arise whenever one quantity depends on another. If your grade on your next math test depends on the number of hours you study, your test grade is a function of the number of hours studied. In mathematics, however, the term function has a narrower meaning. A function is a relationship between two variables, say x and y, such that each possible value of x is associated with exactly one value of y. AN EXAMPLE OF A FUNCTION Let set X consist of five teenagers and set y consist of their possible ages: If each teenager in set X is associated with his or her present age in set Y, the result can be written as a set of ordered pairs: {(Alice, 17), (Barbara, 13), (Chris, 16), (Dennis, 19), (Enid, 15)}. Since each teenager from set X is associated with exactly one age from set Y, the set of ordered pairs is called a function. DEFINITION OF A FUNCTION A function is a set of ordered pairs in which no two ordered pairs of the form (x,y) have the same x -value but different y -values. If f = {(1,1), (2,3), (3,7), (4,5)} then f is a function...
eBook - ePubDifferentiating Instruction in Algebra 1
Ready-to-Use Activities for All Students (Grades 7-10)
- Kelli Jurek(Author)
- 2021(Publication Date)
- Routledge(Publisher)
...Unit 1 IntRoduction to Functions and Relationships DOI: 10.4324/9781003234180-2 Beginning algebra students need to take their understanding of linear equations and solving simple one-variable equations and apply it to studying special relations of data called Functions. Functions describe everyday situations where one specific quantity determines the value of another. Students must learn to write and evaluate Functions because they describe a unique relationship between two quantities and are frequently used to model everyday situations. This unit begins with a preassessment and three real-life applications of Functions that can be discussed in small groups and then as a larger group. Many of the activities will offer the students an opportunity to choose learning activities according to their learning style, personal interests, and readiness level. What Do We Want Students to Know? Common Core State Standards Addressed: • 8.F.1, 4 • A.CED.1, 2 • F.IF.1, 2, 5 • F.BF.1c Big Ideas • Not all relationships are Functions. • A function denotes a special relationship between independent and dependent variables. • A function must pass the vertical line test. Essential Questions • What makes a relationship a function? • How is the vertical line test used to determine if a relationship represents a function? • Does it matter that a set of data does not represent a function? Critical Vocabulary Domain Function notation Dependent variable Output Range Relation Function Independent variable Vertical line test Input Unit Objectives As a result of this unit, students will know: ➤ a function is a special type of relation, ➤ the difference between a relation and a function, ➤ all Functions are relations but not all relations are Functions, ➤ the difference between an independent and dependent variable, and ➤ f x) is read...
eBook - ePub- Iqbal H. Jebril, Hemen Dutta, Ilwoo Cho(Authors)
- 2021(Publication Date)
- CRC Press(Publisher)
...4 Functions DOI: 10.1201/9780429022838-4 The word “function” was first used by G.W. Leibnitz in 1694. J. Bernoulli defined a function as “any expression including variables and constants” in 1698. The familiar notation f (x) was first used by L. Euler in 1734. We shall present in this chapter the basic properties of Functions and induced set Functions. 4.1 Functions 4.1.1 Domain and Range Definition 4.1.1 Let A and B be sets and let f be a relation from A to B. f is said to be a function if the following conditions hold: ∀ x ∈ A, ∃ y ∈ B, such that (x, y) ∈ f. If (x, y) ∈ f and (x, z) ∈ f, then y = z. By the above definition, a relation f is a function from A to B, if it is wholly defined on A, and it has a unique image in B for an arbitrarily fixed element. of A. Example 4.1.2 Let A = { 1, 3, 5, 7 }, B = { 2, 4, 6 }, and let f be a relation defined by f = { (1, 2), (3, 4), (5, 2), (7, 6) }. Then f is a function from A to B. Example. 4.1.3 Let A = { 1, 3, 5, 9 }, B = { 3, 7, 11, 16, 19 }, and let f be a relation from A to B defined by f = { (x, y) : y = 2 x + 1 }. It is clear that f is a function from A to B. Example. 4.1.4 Let A = { 1, 3, 5, 9 }, B = { 3, 7, 11, 16 }, and let f = { (x, y) : y = 2 x + 1 }. Then f is not a function, since 9 ∈ A, but there does not exist y ∈ B, such that (9, y) ∈ f. A function f from A to B is also called a mapping from A to B. We. write f : A → B, and this is read “f maps A to B” or “f is a function from A to B”. The set A is called the domain of f, denoted by Dom (f). The set B is called the codomain of f, denoted by codom (f). Definition 4.1.5 Let f : A → B. If (x, y) ∈ f, then we write y = f (x), and we say that y is the value of f at x, or the image of x under f...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The machine takes in a number, generally uses a rule to perform operations, and puts out another number. There are certain conditions that are set on the machine in order for it to be a function machine. You can call it input and output or x -values and y -values, but there is also new vocabulary. Domain of a function The set of input values of a function. Range of a function The set of output values of a function. Domain input Range output x f (x) x y All the same ideas, just different names. In a function, each element of the domain is paired with exactly one element in the range. x y 1 5 2 7 3 9 4 11 You can represent a relation as a table, graph, or rule (equation), or in words. Functions NOT Functions Think: Does it function? It is okay for the y -value to repeat. For example, y = 2 is a function. It is not okay for the x -values to repeat. For example, x = 2 is not a function. All Functions must pass the vertical line test. The vertical line test is done by drawing a vertical line through the function. If the vertical line has only one point of intersection, then the graph is a function because each input will be paired with one output. Vertical line tests Another difference with Functions is the way you write them. The domain is still written using x ; however, the y is written using f (x). Function notation A notation in which a function is named with a letter and the input is shown in parentheses after the function name. EXAMPLE 12.5 For exercises 1–4, answer the following: What is the domain? What is the range? Is it a function? 1) {(1, 2), (2, 4), (3, 6)} 2) x y 1 5 2 7 3 9 4 11 3) {(0, 2), (2, 4), (0, 5)} 4) x y 1 5 2 3 3 3 4 5 5) Which of the following pass the vertical line test? a) b) c) d) e) f) SOLUTIONS 1) Domain: 1, 2, 3 Range: 2, 4, 6 Function: yes 2) Domain: 1, 2, 3, 4 Range: 5, 7, 9,...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...2 Function Features If a relationship between two real variables, y and x, is such that y becomes determined whenever x is given, then y is said to be a univariate (real‐valued) function of (real‐variable) x ; this is usually denoted as y ≡ y { x }, where x is termed independent variable and y is termed dependent variable. The same value of y may be obtained for more than one value of x, but no more than one value of y is allowed for each value of x. If more than one independent variable exist, say, x 1, x 2, …, x n, then a multivariate function arises, y ≡ y { x 1, x 2, …, x n, }. The range of values of x for which y is defined constitutes its interval of definition, and a function may be represented either by an (explicit or implicit) analytical expression relating y to x (preferred), or instead by its plot on a plane (useful and comprehensive, except when x grows unbounded) – whereas selected values of said function may, for convenience, be listed in tabular form. Among the most useful quantitative relationships, polynomial Functions stand up – of the form P n { x } ≡ a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0, where a 0, a 1, …, a n− 1, and a n denote (constant) real coefficients and n denotes an integer number; a rational function appears as the ratio of two such. polynomials, P n { x }/ Q m { x }, where subscripts n and m denote polynomial degree of numerator and denominator, respectively. Any function y { x } satisfying P { x } y m + Q { x } y m− 1 + ⋯ + U { x } y + V { x } = 0, with m denoting an integer, is said to be algebraic; Functions that cannot be defined in terms of a finite number of said polynomials, say, P { x }, Q { x }, …, U { x }, V { x }, are termed transcendental – as is the case of exponential and logarithmic Functions, as well as trigonometric Functions. A function f is said to be even when f { −x } = f { x } and odd if f { −x } = −f { x }; the vertical axis in a Cartesian...
eBook - ePub- Mel Friedman(Author)
- 2012(Publication Date)
- Research & Education Association(Publisher)
...CHAPTER 5 Functions and Their Graphs CHAPTER 5 Functions AND THEIR GRAPHS ELEMENTARY Functions A function is any process that assigns a single value of y to each number of x. Because the value of x determines the value of y, y is called the dependent variable and x is called the independent variable. The set of all the values of x for which the function is defined is called the domain of the function. The set of corresponding values of y is called the range of the function. PROBLEM Is y 2 = x a function? SOLUTION Graph the equation. Note that x can have two values of y. Therefore, y 2 = x is not a function. PROBLEM Find the domain and range for y = 5 – x 2. SOLUTION First determine if there are any values that would make the function undefined (i.e., division by 0). There are none. Thus, the domain is the set of real numbers. The range can be found by substituting some corresponding values for x in the equation. The range is the set of real numbers less than or equal to 5. PROBLEM Evaluate f (1) for y = f (x) = 5 x + 2. SOLUTION f (x) = 5 x + 2 f (1) = 5(1) + 2 = 5 + 2 = 7 OPERATIONS ON Functions Functions can be added, subtracted, multiplied, or divided to form new Functions. a. (f + g) (x) = f (x) + g (x) b. (f – g) (x) = f (x) – g (x) c. (f × g) (x). = f (x) g (x) d. PROBLEM Let f (x) = 2 x 2 – 1 and g (x) = 5 x + 3. Determine the following Functions: 1. f + g 2. f – g 3. f × g 4. SOLUTION COMPOSITE FUNCTION The composite function f ° g is defined (f ° g)(x) = f (g (x)). PROBLEM Given f (x) = 3 x and g (x) = 4 x + 2. Find (f ° g) (x) and (g ° f) (x). SOLUTION (f ° g) (x) = f (g (x)) =. 3(4 x + 2) = 12 x + 6 (g ° f) (x) = g (f (x)) = 4(3 x) + 2 = 12 x + 2 Note that (f ° g) (x) (g ° f) (x). PROBLEM Find (f ° g) (2) if f (x) = x 2 – 3 and g (x) = 3 x + 1. SOLUTION Substitute the value of x = 2 in g (x): Substitute the value of g (2) in f (x): f (7) = (7) 2 – 3 = 49 – 3 =...
- Jo Boaler, Jen Munson, Cathy Williams(Authors)
- 2019(Publication Date)
- Jossey-Bass(Publisher)
...BIG IDEA 5 The Ins and Outs of Functions Why are Functions and algebra meaningless to so many students? How did Functions, which offer insights into real-world relationships, become a set of rules to remember? US textbooks have led many students, and teachers, astray by introducing algebra and variables to students as a requirement to solve for x. This goes against the research base on algebra in a fundamental way. When we teach students to solve for x, the variable x typically has one solution. But the meaning and essence of variables is that they vary—that they are there to represent a situation that can change. When we show students growing patterns, the case number represents variable, and the case number changes with each case. If students learn that variables can vary, they can later meet situations when the variable has one solution, and this does not provide conflict for them. But when we teach students to solve for x and then ask them to work with situations where the x varies, they become very confused, and some meet a serious conceptual block. This has an easy solution—introduce algebra to students through visual patterns that they can see and understand and represent algebraically. There is another issue with the “solve for x” approach to algebra in textbooks, one that Cornell mathematician Steve Strogatz (2019) discusses in his book Infinite Powers. He describes the algebraic-manipulation approach to mathematics, with variables that are not represented visually, in this way: Equations could be massaged almost mindlessly, peacefully; you could add the same term to both sides of an equation, cancel common terms, solve for an unknown quantity, or perform a dozen other procedures and algorithms according to standard recipes. The processes of algebra could be soothingly repetitive, like the pleasures of knitting. But algebra suffered from its emptiness. Its symbols were vacuous. They meant nothing until they were given meaning. There was nothing to visualize...
eBook - ePub- Thomas Pfaff(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...2 Functions and Their Graphs We define and use Functions throughout this book, and in this chapter we focus on Functions and their associated graphs. We provide examples of commonly used mathematical Functions: piecewise, step, parametric, and polar. We broaden the use of function by using the geometric definition to obtain a parabola, discuss Functions that return Functions, and create a function that returns Pythagorean triples. In each case we graph the function; we keep our graphs basic leaving chapter 3 for further details on graphing, except for our last example where we create a checkerboard graph. The command for creating a function is function() {}, where the variable(s) are listed inside the parenthesis and the function is defined within the braces. In our first example, we define the function f to be x 2 sin (x) and evaluate it at 3 with f(3). There are a number of predefined Functions such as abs, sqrt, the trigonometric Functions, hyperbolic Functions, log for the natural log, log10, log2, and the exponential function exp. So, for example, sin(x) is available to use in our definition of f. Note that * must be used for multiplication as we cannot simply juxtapose objects. R Code > f=function(x){x ^ 2*sin(x)} > f(3) [1] 1.27008 We can plot our function with curve. The first three arguments must be the function, the lower value for the independent variable, and the upper value for the dependent variable. The default range for the dependent variable is selected based on the minimum and maximum of the function on the given interval. There are numerous options, such as ylim for the y limits, lwd for the width (i.e., thickness) of the curve, col for the color of the curve, xlab and ylab for labeling the axis, and lty for the type of line (e.g., dashed, dotted). R Code > curve(f,-5,5) Our next two examples illustrate Functions of two and three variables. The first returns the area of a rectangle given the length and width...
