Functions

Historical View

Function, in mathematics, term used to indicate the relationship or correspondence between two or more quantities. The term function was first used in 1637 by the French mathematician René Descartes to designate a power xⁿ of a variable x. In 1694 the German mathematician Gottfried Wilhelm Leibniz applied the term to various aspects of a curve, such as its slope. The most widely used meaning until quite recently was defined in 1829 by the German mathematician Peter Dirichlet. Dirichlet conceived of a function as a variable y, called the dependent variable, having its values fixed or determined in some definite manner by the values assigned to the independent variable x, or to several independent variables x₁, x₂, ..., x_k.

The values of both the dependent and independent variables were real or complex numbers. The statement y = f(x), read “ y is a function of x,” indicated the interdependence between the variables x and y; f(x) was usually given as an explicit formula, such as f(x) = x² - 3x + 5, or by a rule stated in words, such as f(x) is the first integer larger than x for all x's that are real numbers (see Number). If a is a number, then f(a) is the value of the function for the value x = a. Thus, in the first example, f(3) = 3² - 3 · 3 + 5 = 5, f(-4) = (-4)² - 3(-4) + 5 = 33; in the second example, f(3) = f(3.1) = f(p) = 4.

The emergence of set theory first extended and then altered substantially the concept of a function. The function concept in present-day mathematics may be illustrated as follows. Let X and Y be two sets with arbitrary elements; let the variable x represent a member of the set X, and let the variable y represent a member of the set Y. The elements of these two sets may or may not be numbers, and the elements of X are not necessarily of the same type as those of Y. For example, X might be the set of the 50 states of the United States and Y the set of positive integers. Let P be the set of all possible ordered pairs (x, y) and F a subset of P with the property that if (x₁, y₁) and (x₂, y₂) are two elements of F, then y₁≠y₂ implies that x₁≠x₂—that is, F contains no more than one ordered pair with a given x as its first member. (If x₁≠x₂, however, it may happen that y₁ = y₂.) A function is now regarded as the set F of ordered pairs with the stated condition and is written F:X→Y. The set X₁ of x's that occur as first elements in the ordered pairs of F is called the domain of the function F; the set Y₁ of y's that occur as second elements in the ordered pairs is called the range of the function F. Thus, {(New York, 7), (Ohio, 4), (Utah, 4)} is one function that has X = the set of the 50 U.S. states and Y = the set of all positive integers; the domain is the three states named, and the range is 4, 7.

The modern concept of a function is related to the Dirichlet concept. Dirichlet regarded y = x² - 3x + 5 as a function; today, y = x² - 3x + 5 is thought of as the rule that determines the correspondent y for a given x of an ordered pair of the function; thus, the preceding rule determines (3, 5), (-4, 33) as two of the infinite number of elements of the function. Although y = f(x) is still used today, it is better to read it as “y is functionally related to x”.

A function is also called a transformation or mapping in many branches of mathematics. If the range Y₁ is a proper subset of Y (that is, at least one y is in Y but not in Y₁), then F is a function or transformation or mapping of the domain X₁ into Y; if Y₁ = Y,F is a function or transformation or mapping of X₁ onto Y.

Concept

• Graphing a Linear Function of the Form f(x) = ax
• Graphing a Linear Function of the Form f(x) = ax + b
• Defining a Linear Function of the Form f(x) = ax
• Defining a Linear Function of the Form f(x) = ax + b
• Recognizing a Linear Function of the Form f(x) = ax
• Recognizing a Linear Function of the Form f(x) = ax + b
• Square and Reciprocal Functions
• Graphing Functions

Inverse Functions: function that exactly reverses the transformation produced by a function f. It is usually written as f^–1. For example 3x + 2 and (x—2)/3 are mutually inverse functions. Multiplication and division are inverse operations.
Example:
Q. Find the inverse of function f(x)=3x+2:

Answer:

• To find the inverse of the function, interchange the variables and solve for f^(-1)(x).
• x=3f^–1(x)+2
• Since f^(-1)(x) is on the right-hand side of the equation, switch the sides so it is on the left-hand side of the equation.
• 3f^–1(x)+2=x
• Since 2 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 2 from both sides.
• 3f^–1(x)=-2+x
• Move all terms not containing f^(-1)(x) to the right-hand side of the equation.
• 3f^–1(x)=x-2
• Divide each term in the equation by 3.
• (3f^–1(x))/(3)=(x)/(3)-(2)/(3)
• Simplify the left-hand side of the equation by canceling the common factors.
• f^–1(x)=(x)/(3)-(2)/(3)
• Combine the numerators of all expressions that have common denominators.
• f^–1(x)=(x-2)/(3)