FUNCTIONS
Def. Variable. A symbol, such as x, that can stand for any one of a set of numbers. Any member of the set is a value of the variable and the set itself is the range of the variable. A symbol, such as z, that can stand for any one of a set of complex numbers is called a complex variable.
Letters at the end of the alphabet such as x, y, z, u, v, w, are usually employed to represent variables and letters at the beginning of the alphabet are usually used for constants.
Def. Function (of a variable). A variable y is said to be a function of a variable x, written y = f(x), if for each of a certain set of values of x there is a corresponding value of y. The variable x is called the independent variable and the variable y is called the dependent variable. The value of a function y = f(x) at x = a is written f(a).
The concept of a function y = f(x) is most easily understood by considering its graph in an Cartesian coordinate system. In Fig. 1 a variable y is depicted as a function of a variable x. The variable x is the independent variable and the variable y is the dependent variable. Now consider Fig. 2. As x progresses from point a to b on the x axis, passing through successive values x1, x2, x3, ... , xn, the values of y = f(x) progress through values ξ1, ξ2, ξ3, ... ,ξn. The central idea is that as the independent variable x ranges over some sequence of values from some set (i.e. values on some interval [a, b] of the x axis), the dependent variable y varies, assuming a succession of values.
Examples of functions.
1] The formula for the area of a circle, A = πr2, gives the area A of a circle as a function of the radius r. To each value of r there corresponds a value for A.
2] The equation
y = 5x2 + 3x + 2
is a function. It gives a value of y for each value of x in the interval consisting of the entire x axis.
● In general, any equation or formula that gives the value of one variable in terms of another represents a function.
3] A chart showing the air temperature at some point in space as measured over some period of time defines a function of temperature v.s. time for that location.
● We see from this last example that a function need not be defined by some formula or analytic expression. It can be defined by some chart, graph, or table.
4] Let [x] represent the integral part of a real number i.e. the integer obtained by truncating the fractional part. For example, [2.1] = 2, [5.4] = 5 , etc. Then y = [x] is the function shown in Fig. 3.
5] The function w = 5z2 + 2z + 1, where z is a complex number, supplies to any specified value of z, a value of w i.e. it assigns a complex number w to the complex number z.
Domain of definition of a function. In general, the independent variable is defined over some set of values. There are frequently restrictions, either implied or stated, on the values that may be assumed by the independent variable. The set of values over which the independent variable is defined is called the domain of definition of the function.
Example. In the case of the formula for the area of a circle, A = πr2, the range of definition for the independent variable r is the positive numbers. A negative value of r makes no sense.
In the definition above we have defined a function for the case of a single independent variable. There can, however, be more than one independent variable. For example, the equation z = 3x2 + 5 sin y + 2 represents such a function. It defines the variable z as a function of independent variables x and y. Fig. 4 shows a function z = f(x, y) where x and y are the independent variables
and z is the dependent variable. For each pair (xi, yi) in some delineated portion of the x-y plane there corresponds a zi. In the general case there may be n independent variables giving functions of the type
y = f(x1, x2, ... , xn) .
In this case numerical values are assigned to n-tuples i.e. to points of n-space. The domain of definition of such a function corresponds to that set of points in n-space defined by the ranges (i.e. intervals [ai, bi] ) over which the variables x1, x2, ... , xn are allowed to run. The function f assigns a value to each point in the domain.
Functional notation. Symbols such as f, F, g, G, Φ, etc., are used to denote a function, the function values corresponding to x being denoted by f(x), F(x), g(x), G(x), Φ(x), etc. and read as "f of x" or "the f function of x," etc.
Single and multiple-valued functions. One does encounter in mathematics what are called multiple-valued functions. For example, if we solve the equation y2 - x2 = 1 we obtain
This is an example of what is called a multiple-valued function. If, in a function y = f(x), only one value of y corresponds to each value of x, the function is said to be single-valued. If more than one value of y corresponds to a single value of x, the function is said to be multiple-valued. A multiple-valued function can be viewed as a collection of single-valued functions with each of the single-valued functions regarded as a branch of the multiple-valued function.
Example 1. If we solve the equation y2 - x2 = 1 we obtain
This function is multiple-valued with the two branches
Example 2. The function
where z is a complex number, is a multiple-valued function with five branches, one branch for each of the five roots of z.
Often one of the branches of a multiple-valued function is selected and regarded as the principal branch.
Generally speaking, multiple-valued functions are regarded as collections of single-valued functions and when we speak of functions we assume them to be single-valued. Two central concepts in mathematics are the concepts of the derivative and integral. Both of these concepts presume a function that is single-valued. Whenever we speak of a derivative or an integral of a function, we tacitly assume the function is single-valued. Some authors, in defining a function, allow a function to have multiple values. Other authors require it to have only one value. Thus there are two different conflicting definitions that are used for the word “function”.
Inverse functions. If y = f(x), then we can generally, for some interval of interest, also view x as a function of y, x = g(y), or x = f -1(y). The function x = g(y), or x = f -1(y), is called the inverse of the function y = f(x).
A function has an inverse if and only if it is one-to-one.
Expressions viewed as functions. Let E(x1, x2, ... ,xn) be an arbitrary expression (algebraic, transcendental, etc.) in n variables. Then y = E(x1, x2, ... ,xn) is a function which assigns a number to each set of values of x1, x2, ... ,xn in n-space. It is an important function associated with the expression E(x1, x2, ... ,xn). The domain is the set of all n-tuples i.e. all points in n-space. The range is some set of real or complex numbers. In the case when n = 1 we have expressions such as 3x2 + 5 sin 2x + log x, 5x5 + 1, etc. They correspond to functions of the type y = f(x) which assign some number to each value of x. In the case when n = 2 we have functions of the type y = f(x, y) which assign some number to each pair of numbers (x, y). In the case when n = 3 we have functions of the type y = f(x, y, z) which assign some number to each triplet (x, y, z).
Some comments. When we talk about functions we will typically say things like the function y = x2 + 5 . This is really incorrect because an equation is not a function. We should say the function defined by y = x2 + 5 to be technically correct. Language can be confusing. We may think of the equation defining a function as the function, or the graphical representation of the function as the function, but one is an equation and the other is a graphical representation. Functions are defined by expressions. Any expression can be viewed as defining a function. The variables contained in the expression are the independent variables and the value of the expression is the dependent variable.
As a student progresses through the subjects of algebra, analytic geometry, calculus, etc. the functions he invariably encounters are defined by algebraic or transcendental expressions. He becomes conditioned over the years to thinking of a function as something defined by some expression i.e. defined by “y = some expression”. As he gets into more advanced mathematics, however, he may encounter an author using the term in a broader way to include functions that are not represented by expressions and may find himself confused. He may rebel at the idea of using the term function for varying quantities that cannot be expressed in algebraic terms. He may think, “Why include those under the term function? You can’t manipulate them algebraically. They are a totally different animal from the usual concept of a function. You should call them something else.” In fact, the term function is used to include varying quantities that may not be expressible in algebraic terms. We may talk about the temperature, density, velocity, etc. in some medium as a function of, say, time even though we don’t know what algebraic expression might describe it.