Function
Function
A function represents a mathematical relationship between two sets of real numbers. These sets of numbers are related to each other by a rule that assigns each value from one set to exactly one value in the other set. The standard notation for a function y = f (x), developed in the eighteenth century, is read ‘y equals f of x.’ Other representations of functions include graphs and tables. The types of rules that govern their relationships classify functions. Some of these classifications include algebraic, trigonometric, and logarithmic and exponential. It has been found by mathematicians and scientists alike, that these elementary functions can represent many real-world phenomena. In computer science, functions are used to represent data structures and to describe algorithms.
History of functions
The idea of a function was developed in the seventeenth century. During this time, French mathematician and philosopher Rene Descartes (1596–1650), in his book Geometry (1637), used the concept to describe many mathematical relationships. The term function was introduced by German mathematician Gottfried Wilhelm Leibniz (1646–1716) almost 50 years after the publication of Geometry. The idea of a function was further formalized by Swiss mathematician Leonhard Euler (1707–1783; pronounced “oiler”) who introduced the notation for a function, y = f (x).
Characteristics of functions
The idea of a function is very important in mathematics because it describes any situation in which one quantity depends on another. For example, the distance an object travels in four hours depends on its speed. When such relationships exist, one variable is said to be a function of the other. Therefore, height is a function of age and distance is a function of speed.
The relationship between the two sets of numbers of a function can be represented by a mathematical equation. Consider the relationship of the area of a square to its sides. This relationship is expressed by the equation A = x2. Here, A, the value for the area, depends on x, the length of a side. Consequently, A is called the dependent variable and x is the independent variable. In fact, for a relationship between two variables to be called a function, every value of the independent variable must correspond to exactly one value of the dependent variable.
The previous equation mathematically describes the relationship between a side of the square and its area. In functional notation, the relationship between any square and its area could be represented by f(x) = x2, where A = f(x). To use this notation, one substitutes the value found between the parentheses into the equation. For a square with a side 4 units long, the function of the area is f(4) = 42, or 16. Using f(x) to describe the function is a matter of tradition. However, one could use almost any combination of letters to represent a function such as g(s), p(q), or even LMN(z).
The set of numbers made up of all the possible values for x is called the domain of the function. The set of numbers created by substituting every value for x into the equation is known as the range of the function. For the function of the area of a square, the domain and the range are both the set of all positive real numbers. This type of function is called a one-to-one function because for every value of x, there is one and only one value of A. Other functions are not one-to-one because there are instances when two or more independent variables correspond to the same dependent variable. An example of this type of function is f(x) = x2. Here, f(2) = 4 and f(-2) = 4.
Just as one added, subtracted, multiplied, or divided real numbers to get new numbers, functions can be manipulated as such to form new functions. Consider the functions f (x) = x2 and g(x) = 4x + 2. The sum of these functions f (x) +g(x) =x2 + 4x + 2. The difference of f (x) - g(x) = x2 - 4x - 2. The product and quotient can be obtained in a similar way. A composite function is the result of another manipulation of two functions. The composite function created by our previous example is noted by f(g(x)) and equal to f(4x + 2) = (4x + 2)2. It is important to note that this composite function is not equal to the function g(f(x)).
Functions which are one-to-one have an inverse function which will undo the operation of the original function. The function f(x) = x + 6 has an inverse function denoted as f-1(x) = x - 6. In the original function, the value for f(5) = 5 + 6=11. The inverse function reverses the operation of the first so, f-1(11) = 11 - 6 = 5.
In addition to a mathematical equation, graphs and tables are another way to represent a function. Since a function is made up of two sets of numbers each of which is paired with only one other number, a graph of a function can be made by plotting each pair on an X,Y coordinate system known as the Cartesian coordinate system. Graphs are helpful because they allow one to visualize the relationship between the domain and the range of the function.
Classification of functions
Functions are classified by the type of mathematical equation that represents their relationship. Some functions are algebraic. Other functions like f(x) = sin x, deal with angles and are known as trigonometric. Still other functions have logarithmic and exponential relationships and are classified as such.
Algebraic functions are the most common type of function. These are functions that can be defined using addition, subtraction, multiplication, division, powers, and roots. For example f(x) = x + 4 is an algebraic function, as is f(x) = x/2 or f(x) = x3 . Algebraic functions are called polynomial functions if the equation involves powers of x and constants. The most famous of these is the quadratic function (quadratic equation), f(x) = ax2 + bx + c, where a, b, and c are constant numbers.
A type of function that is especially important in geometry is the trigonometric function. Common trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent. One interesting characteristic of trigonometric functions is that they are periodic. This means there are an infinite number of values of x that correspond to the same value of the function. For the function f(x) = cos x, the x values 90° and 270° both give a value of 0, as do 90° + 360° = 450° and 270° + 360° = 630°. The value 360° is the period of the function. If p is the period, then f(x + p) = f(x) for all x.
Exponential functions can be defined by the equation f(x) = bx, where b is any positive number except 1. The variable b is constant and known as the base. The most widely used base is an irrational number denoted by the letter e, which is approximately equal to 2.7182818284. Exponential functions, along with the mathematical constant e, are used in complex equations to describe growth or decay processes.
Key Terms
Dependent variable— The variable in a function whose value depends on the value of another variable in the function.
Independent variable— The variable in a function that determines the final value of the function.
Inverse function— A function that reverses the operation of the original function.
One-to-one function— A function in which there is only one value of x for every value of y and one value of y for every x.
Range— The set containing all the values of the function.
Logarithmic functions are the inverse of exponential functions. For the exponential function y = 4x, the logarithmic function is its inverse, x = 4y and would be denoted by y = f(x) = log4 x. Logarithmic functions having a base of e are known as natural logarithms and use the notation f(x) = ln x.
Mathematicians and other scientists use functions in a wide variety of areas to describe and predict natural events. Chemists and physicists use algebraic functions extensively. Trigonometric functions are particularly important in architecture, astronomy, and navigation. Financial institutions use exponential and logarithmic functions. In each case, the power of the function allows people to take mathematical ideas and apply them to real world situations.
Resources
BOOKS
Burton, David M. The History of Mathematics: An Introduction. New York: McGraw-Hill, 2007.
Jeffrey, Alan. Mathematics for Engineers and Scientists. Boca Raton, FL: Chapman & Hall/CRC, 2005.
Larson, Ron. Calculus: An Applied Approach. Boston, MA: Houghton Mifflin, 2003.
_____. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.
Lorenz, Falko. Algebra. New York: Springer, 2006.
Lyublinskava, Irina E. Connecting Mathematics with Science: Experiments for Precalculus. Emeryville, CA: Key Curriculum Press, 2003.
Setek, William M. Fundamentals of Mathematics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.
Perry Romanowski
function
There has been a strong and often explicit functionalism present in sociology and social anthropology throughout most of this century. There has also been an implicit functionalism in the more determinist forms of Marxist theory, where so-called surface features of the social formation (such as political systems, ideologies, and trade unions), are seen as produced by, in order to maintain, the underlying relations of production. However, probably the most famous functionalist analysis in sociology is the so-called functional theory of social stratification offered by Kingsley Davis and Wilbert Moore, although Davis also wrote a functionalist textbook, Human Society (1949), and made a spirited defence of functionalism in his Presidential Address to the American Sociological Association in 1959 (see ‘The Myth of Functional Analysis as a Special Method in Sociology and Anthropology’, American Sociological Review, 1959
). Herbert J. Gans 's celebrated essay on ‘The Positive Functions of Poverty’ (American Journal of Sociology, 1972)
, said by some to have been written as a parody of structural-functionalism, is actually a superb example of ideologically neutral functional analysis.
In the late 1960s functionalism came under sustained attack from various sources. It was argued that this approach could not account for social change, or for structural contradictions and conflict in societies, and that its reliance on stability and on the organic analogy rendered it ideologically conservative: it became fashionable to refer to functionalism as consensus theory. This particular group of criticisms is not entirely accurate. Parsons's evolutionary theory, seeing historical development in terms of the differentiation and reintegration of systems and subsystems, can account for change and at least for temporary conflict until the reintegration takes place. The existence of functional explanations in Marxism indicates that they can exist alongside a recognition of contradictions in social systems. Durkheim himself was able to combine functionalist explanations with a sometimes radical form of guild socialism.
The telling criticisms of functionalism have been epistemological and ontological. The epistemological argument is that a functionalist explanation is not an explanation at all in that it does not identify causal mechanisms and processes; it is, instead, assumed that social institutions are adequately explained in terms of their putative effects. The ontological arguments have to do with what we think is the nature of society itself. Some theorists, who are happy to accept that society has an existence over and above individuals, nevertheless also argue that we cannot attribute needs (for example Parsons's four, famous, so-called functional prerequisites of adaptation, goal attainment, integration, and latency) to a society as such, since that is to grant societies the same qualities as human beings. Furthermore, even if we can attribute needs to a society, it does not follow that because these needs exist they will be met. It requires a proper historical and causal explanation to show why and how they are met. Anthony Giddens argues that all functionalist explanations can be rewritten as historical accounts of human action and its consequences; that is, human individuals and their actions are the only reality, and we cannot regard societies or systems as having an existence over and above individuals.
For most of the 1970s and a good part of the 1980s, it seemed as if functionalism as a school of thought and as a way of understanding and explaining social phenomena had disappeared, but during recent years there have been some interesting attempts at a revival: in America under the impetus of Jeffrey Alexander; in Germany in the work of Niklas Luhman; and, in Britain, in an interesting revision of Marxism by the philosopher G. A. Cohen.
Alexander argues (in Neofunctionalism, 1985) that functionalism is perhaps best understood as a broad school (rather like Marxism), in which there are many variations of approach, rather than a systematic theory in the manner of Parsons. He maintains that we should not take it as providing explanations, but as a description which focuses on the symbiotic relationships between social institutions and their environment, taking equilibrium (stability) as a reference-point for analysis, rather than as something which necessarily exists in reality, and treating structural differentiation as a major form of social change. This effectively strips functionalism of the determinism of systems theory. For Alexander, functionalism is simply one approach among many, and has the virtue of focusing attention on aspects of the social ignored elsewhere.
Cohen's argument (in Inquiry, 1982) takes up a position which can be found in a different form in Durkheim's work. He suggests that societies can be seen, not as having needs in the way that individuals can be said to have needs, but as having what he calls dispositional facts; that is, features of a social environment which encourage the continued existence of a particular institution, but did not actually cause that institution to come into existence. Cohen's example is racism, which historically might be the result of a range of factors, but which survives because once in existence it helps the capitalist system to survive, by dividing the working class and making social control easier. In a rather similar way Jon Elster, a leading exponent of modern rational-choice theory, argues that we have to employ a functionalist explanation to show why capitalist firms on average adopt a policy of profit maximization. Independently of how they come into existence, the market selects for survival those that come closest to this optimal strategy, and thus imposes it upon them (see Ulysses and the Sirens, 1979
). Functionalism, then, still has a place in sociology–albeit a more restricted place than when the Parsonsian version was dominant. See also DEVIANCE, SOCIOLOGY OF; DEVELOPMENT, SOCIOLOGY OF; DIVISION OF LABOUR; MALINOWSKI, BRONISLAW; RADCLIFFE-BROWN, A. R.; SOCIAL INTEGRATION AND SYSTEM INTEGRATION; SYSTEMS THEORY.
Function
Function
A function represents a mathematical relationship between two sets of real numbers . These sets of numbers are related to each other by a rule which assigns each value from one set to exactly one value in the other set. The standard notation for a function y = f(x), developed in the 18th century, is read "y equals f of x." Other representations of functions include graphs and tables. Functions are classified by the types of rules which govern their relationships including; algebraic, trigonometric, and logarithmic and exponential. It has been found by mathematicians and scientists alike that these elementary functions can represent many real-world phenomena.
History of functions
The idea of a function was developed in the seventeenth century. During this time, Rene Descartes (1596-1650), in his book Geometry (1637), used the concept to describe many mathematical relationships. The term "function" was introduced by Gottfried Wilhelm Leibniz (1646-1716) almost fifty years after the publication of Geometry. The idea of a function was further formalized by Leonhard Euler (pronounced "oiler" 1707-1783) who introduced the notation for a function, y = f(x).
Characteristics of functions
The idea of a function is very important in mathematics because it describes any situation in which one quantity depends on another. For example, the height of a person depends on his age. The distance an object travels in four hours depends on its speed. When such relationships exist, one variable is said to be a function of the other. Therefore, height is a function of age and distance is a function of speed.
The relationship between the two sets of numbers of a function can be represented by a mathematical equation. Consider the relationship of the area of a square to its sides. This relationship is expressed by the equation A = x2. Here, A, the value for the area, depends on x, the length of a side. Consequently, A is called the dependent variable and x is the independent variable. In fact, for a relationship between two variables to be called a function, every value of the independent variable must correspond to exactly one value of the dependent variable.
The previous equation mathematically describes the relationship between a side of the square and its area. In functional notation, the relationship between any square and its area could be represented by f(x) = x2, where A = f(x). To use this notation, we substitute the value found between the parenthesis into the equation. For a square with a side 4 units long, the function of the area is f(4) = 42 or 16. Using f(x) to describe the function is a matter of tradition. However, we could use almost any combination of letters to represent a function such as g(s), p(q), or even LMN(z).
The set of numbers made up of all the possible values for x is called the domain of the function. The set of numbers created by substituting every value for x into the equation is known as the range of the function. For the function of the area of a square, the domain and the range are both the set of all positive real numbers. This type of function is called a one-to-one function because for every value of x, there is one and only one value of A. Other functions are not one-to-one because there are instances when two or more independent variables correspond to the same dependent variable. An example of this type of function is f(x) = x2. Here, f(2) = 4 and f(-2) = 4.
Just as we add, subtract, multiply or divide real numbers to get new numbers, functions can be manipulated as such to form new functions. Consider the functions f(x) = x2 and g(x) = 4x + 2. The sum of these functions f(x) + g(x) = x2 + 4x + 2. The difference of f(x) - g(x) = x2 - 4x - 2. The product and quotient can be obtained in a similar way. A composite function is the result of another manipulation of two functions. The composite function created by our previous example is noted by f(g(x)) and equal to f(4x + 2) = (4x + 2)2. It is important to note that this composite function is not equal to the function g(f(x)).
Functions which are one-to-one have an inverse function which will "undo" the operation of the original function. The function f(x) = x + 6 has an inverse function denoted as f-1(x) = x - 6. In the original function, the value for f(5) = 5 + 6 = 11. The inverse function reverses the operation of the first so, f-1(11) = 11 - 6 = 5.
In addition to a mathematical equation, graphs and tables are another way to represent a function. Since a function is made up of two sets of numbers each of which is paired with only one other number, a graph of a function can be made by plotting each pair on an X,Y coordinate system known as the Cartesian coordinate system. Graphs are helpful because they allow you to visualize the relationship between the domain and the range of the function.
Classification of functions
Functions are classified by the type of mathematical equation which represents their relationship. Some functions are algebraic. Other functions like f(x) = sin x, deal with angles and are known as trigonometric. Still other functions have logarithmic and exponential relationships and are classified as such.
Algebraic functions are the most common type of function. These are functions that can be defined using addition, subtraction , multiplication , division , powers, and roots. For example f(x) = x + 4 is an algebraic function, as is f(x) = x/2 or f(x) = x3. Algebraic functions are called polynomial functions if the equation involves powers of x and constants. The most famous of these is the quadratic function (quadratic equation), f(x) = ax2 + bx + c where a, b, and c are constant numbers.
A type of function that is especially important in geometry is the trigonometric function. Common trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent. One interesting characteristic of trigonometric functions is that they are periodic. This means there are an infinite number of values of x which correspond to the same value of the function. For the function f(x) = cos x, the x values 90° and 270° both give a value of 0, as do 90° + 360° = 450° and 270° + 360° = 630°. The value 360° is the period of the function. If p is the period, then f(x + p) = f(x) for all x.
Exponential functions can be defined by the equation f(x) = bx, where b is any positive number except 1. The variable b is constant and known as the base. The most widely used base is an irrational number denoted by the letter e, which is approximately equal to 2.71828183. Logarithmic functions are the inverse of exponential functions. For the exponential function y = 4x, the logarithmic function is its inverse, x = 4y and would be denoted by y = f(x) = log4 x. Logarithmic functions having a base of e are known as natural logarithms and use the notation f(x) = ln x.
We use functions in a wide variety of areas to describe and predict natural events. Algebraic functions are used extensively by chemists and physicists. Trigonometric functions are particularly important in architecture, astronomy , and navigation. Financial institutions use exponential and logarithmic functions. In each case, the power of the function allows us to take mathematical ideas and apply them to real world situations.
Resources
books
Kline, Morris. Mathematics for the Nonmathematician. New York: Dover Publications, 1967.
Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.
Paulos, John Allen. Beyond Numeracy. New York: Alfred A. Knopf Inc., 1991.
Perry Romanowski
KEY TERMS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .- Dependent variable
—The variable in a function whose value depends on the value of another variable in the function.
- Independent variable
—The variable in a function which determines the final value of the function.
- Inverse function
—A function which reverses the operation of the original function.
- One-to-one function
—A function in which there is only one value of x for every value of y and one value of y for every x.
- Range
—The set containing all the values of the function.
Function
Function
A function is a mathematical relationship between two sets of real numbers. These sets of numbers are related to each other by a rule that assigns each value from one set to exactly one value in the other set. For example, suppose we choose the letter x to stand for the numbers in one set and the letter y for the numbers in the second set. Then, for each value we assign to x, we can find one and only one comparable value of y.
An example of a function is the mathematical equation y = 3x + 2. For any given value of x, there is one and only one value of y. If we choose 5 for the value of x, then y must be equal to 17 (3 · 5 + 2 = 17). Or if we choose 11 for the value of x, then y must be equal to 35 (3 · 11 + 2 = 35).
The standard notation for a function is y = f(x) and is read "y equals f of x." Functions can also be represented in other ways, such as by graphs and tables. Functions are classified by the types of rules that govern their relationships: algebraic, trigonometric, logarithmic, and exponential. Mathematicians and scientists have found that elementary functions represent many real-world phenomena.
Characteristics of functions
The idea of a function is very important in mathematics because it describes any situation in which one quantity depends on another. For example, the height of a person depends, to a certain extent, on that person's age. The distance an object travels in four hours depends on its speed. When such relationships exist, one variable is said to be a function of the other. Therefore, height is a function of age and distance is a function of speed.
One way to represent the relationship between the two sets of numbers of a function is with a mathematical equation. Consider the relationship of the area of a square to its sides. This relationship is expressed by the equation A = x2. Here, A, the value for the area, depends on x, the length of a side. Consequently, A is called the dependent variable and x is the independent variable. In fact, for a relationship between two variables to be called a function, every value of the independent variable must correspond to exactly one value of the dependent variable.
The previous equation mathematically describes the relationship between a side of the square and its area. In functional notation, the relationship between any square and its area could be represented by f(x) = x2, where A = f(x). To use this notation, we substitute the value found between the parentheses into the equation. For a square with a side 4 units long, the function of the area is f(4) = 42 or 16. Using f(x) to describe the function is a matter of tradition. However, we could use almost any combination of letters to represent a function such as g(s), p(q), or even LMN(z).
Words to Know
Dependent variable: The variable in a function whose value depends on the value of another variable in the function.
Independent variable: The variable in a function that determines the final value of the function.
Inverse function: A function that reverses the operation of the original function.
Just as we add, subtract, multiply, or divide real numbers to get new numbers, functions can be manipulated as such to form new functions. Consider the functions f(x) = x2 and g(x) = 4x + 2. The sum of these functions f(x) + g(x) = x2 + 4x + 2. The difference of f(x) − g(x) = x2 − 4x + 2. The product and quotient can be obtained in a similar way.
In addition to a mathematical equation, graphs and tables can be used to represent a function. Since a function is made up of two sets of numbers—each of which is paired with only one other number—a graph of a function can be made by plotting each pair on an x, y coordinate system known as the Cartesian coordinate system. Graphs are helpful because they make it easier to visualize the relationship between the domain and the range of the function.
Classification of functions
Functions are classified by the type of mathematical equation that represents their relationship. Algebraic functions are the most common type of function. These are functions that can be defined using addition, subtraction, multiplication, division, powers, and roots. Examples of algebraic functions include the following: f(x) = x + 4 and f(x) = x/2 and f(x) = x3.
Two other common types of functions are trigonometric and exponential (or logarithmic) functions. Trigonometric functions deal with the sizes of angles and include the functions known as the sine, cosine, tangent, secant, cosecant, and cotangent. Exponential functions can be defined by the equation f(x) = bx, where b is any positive number except 1. The variable b is constant and is known as the base.
An example of an exponential function is f(x) = 10x. Notice that for values of x equal to 1, 2, 3, and 4, the values of f(x) are 10, 100, 1,000, and 10,000. One property of exponential functions is that they change very rapidly with changes in the independent variable.
The inverse of an exponential function is a logarithmic function. In the equation f(x) = 10x, one procedure is to set certain values of x (as we did in the example above) and then find the corresponding values of f(x). Another possibility is to set certain values of f(x) and find out what values of x are needed to produce those values. This process is using the exponential function in reverse and is known as a logarithmic function.
Applications
All types of functions have many practical applications. Algebraic functions are used extensively by chemists and physicists. Trigonometric functions are particularly important in architecture, astronomy, and navigation. Financial institutions often use exponential and logarithmic functions.
function
1. from one set X to another set Y. A relation R defined on the Cartesian product x × y in which for each element x in X there is precisely one element y in Y with the property that (x,y) is a member of R. It is then customary to talk about a function f, say, and to write f : X → Y
The unique association between elements x and y is denoted by y = f(x) or y = fx
X is called the domain of f, Y the codomain of f. Further, y is the value of f at the point x or the image of x under f. We say that f is a mapping or transformation between sets X and Y or that f maps X into Y, and that f maps x into y. When the domain X is the Cartesian product of n sets then f is a function of n variables. Otherwise it is a function of one variable.
Examples of functions are readily obtained from the mathematical equivalents of standard functions and operations typically supplied in programming languages. The usual trigonometric functions sin, cos, and tan are functions of one variable. The rule for converting from characters into their integer codes or equivalents is a function.
Functions are often represented pictorially as graphs.
See also bijection, injection, surjection, operation, homomorphism.
2. A program unit that given values for input parameters computes a value. Examples include the standard functions such as sin(x), cos(x), exp(x); in addition most languages permit user-defined functions. A function is a “black box” that can be used without any knowledge or understanding of the detail of its internal working. In some languages a function may have side effects.
function
func·tion / ˈfəngkshən/ • n. 1. an activity or purpose natural to or intended for a person or thing: bridges perform the function of providing access across water| Vitamin A is required for good eye function. ∎ practical use or purpose in design: building designs that prioritize style over function. ∎ a basic task of a computer, esp. one that corresponds to a single instruction from the user.2. Math. a relationship or expression involving one or more variables: the function (bx + c). ∎ a variable quantity regarded in relation to one or more other variables in terms of which it may be expressed or on which its value depends. ∎ Chem. a functional group.3. a thing dependent on another factor or factors: class shame is a function of social power.4. a large or formal social event or ceremony: he was obliged to attend party functions.• v. [intr.] work or operate in a proper or particular way: her liver is functioning normally. ∎ (function as) fulfill the purpose or task of (a specified thing): the museum intends to function as an educational and study center.DERIVATIVES: func·tion·less adj.
function
function
Hence vb. XIX. functional XVII, functionary XVIII.