Functions

Suppose that a particle is moving in the space. We assume the physical particle as a point. As time varies, the particle changes its position. Mathematically at any time the point occupies a position in the three dimensional space R .

Let us assume that the time varies from to . So the movement or functioning of the particle decides the position of the particle at any given time t between and . In other words, for each t ∈ [ , ] , the functioning of the particle gives a point in R . Let us denote the position of the particle at time t as f ( t ) .

Let us see another simple example. We know that the equation x − y = describes a straight line. Here whenever x assumes a value, y assumes some value accordingly. The movement or functioning of y is decided by that of x .

Let us denote y by f ( x ) . We may see many situation like this in nature. In the study of natural phenomena, we find that it is necessary to consider the variation of one quantity depending on the variation of another. The relation of the time and the position of the particle, the relation of a point in the x -axis to a point in the y -axis and many more such relations are studied for a very long period in the name function .

Before Cantor, the term function is defined as a rule which associates a variable with another variable. After the development of the concept of sets, a function is defined as a rule that associates for every element in a set A , a unique element in a set B . However the terms rule and associate are not properly defined mathematical terminologies. In modern mathematics every term we use has to be defined properly.

So a definition for function is given using relations. Suppose that we want to discuss a test written by a set of students. We shall see this as a relation. Let A be the set of students appeared for an examination and let B = { , , , , .

. . } be the set of possible marks. We define a relation R as follows: A student a is related to a mark b if a got b marks in the test.

We observe the following from this example: • Every student got a mark. In other words, for every a ∈ A , there is an element b ∈ B such that ( a, b ) ∈ R . • A student cannot get two different marks in any test. In other words, for every a ∈ A , there is definitely only one b ∈ B such that ( a, b ) ∈ R .

This can be restated in a different way: If ( a, b ) , ( a, c ) ∈ R then b = c . Relations having the above two properties form a very important class of relations, called functions. Let us now have a rigorous definition of a function through relations. Definition .

Let A and B be two sets. A relation f from A to B , a subset of A × B , is called a function from A to B if it satisfies the following: (i) for all a ∈ A , there is an element b ∈ B such that ( a, b ) ∈ f . (ii) if ( a, b ) ∈ f and ( a, c ) ∈ f then b = c . That is, a function is a relation in which each element in the domain is mapped to exactly one element in the range.

A is called the domain of f and B is called the co-domain of f . If ( a, b ) is in f , then we write f ( a ) = b ; the element b is called the image of a and the element a is called a pre-image of b and f ( a ) is known as the value of f at a . The set { b : ( a, b ) ∈ f for some a ∈ A } is called the range of the function. If B is a subset of R , then we say that the function is a real-valued function .

Sets, Relations and Functions Two functions f and g are said to be equal functions if their domains are same and f ( a ) = g ( a ) for all a in the domain. If f is a function with domain A and co-domain B , we write f : A → B (Read this as f is from A to B or f be a function from A to B ). We also say that f maps A into B . If f ( a ) = b , then we say f maps a to b or a is mapped onto b by f , and so on.

The range of a function is the collection of all elements in the co-domain which have pre-images. Clearly the range of a function is a subset of the co-domain. Further the first condition says that every element in the domain must have an image; this is the reason for defining the domain of a relation R from a set A to a set B as the set of all elements of A having images and not as A . The second condition says that an element in the domain cannot have two or more images.

Naturally one may have the following doubts: • In the definition, why we use the definite article “the” for image of a and the indefinite article “ a ” for pre-image of b ? • We have a condition stating that every element in the domain must have an image; is there any condition like “every element in the co-domain must have a pre-image”? If not, why? • We have a condition stating that an element in the domain cannot have two or more images; is there any condition like “an element in the co-domain cannot have two or more pre-images”?

If not, why? As an element in the domain has exactly one image and an element in the co-domain can have more than one pre-image according to the definition, we use the definite article “the” for image of a and the indefinite article “ a ” for pre-image of b . There are no conditions as asked in the other two questions; the reason behind it can be understood from the problem of students’ mark we considered above. We observe that every function is a relation but a relation need not be a function.

Let f = { ( a, ) , ( b, ) , ( c, ) , ( d, ) } . Is f a function? This is a function from the set { a, b, c, d } to { , , } . This is not a function from { a, b, c, d, e } to { , , , } because e has no image.

This is not a function from { a, b, c, d } to { , , , } because the image of d is not in the co-domain; f is not a subset of { a, b, c, d }×{ , , , } . So whenever we consider a function the domain and the co-domain must be stated explicitly. The relation discussed in Illustration . is a function with domain { L, E, T, U, S, W, I, N } and co-domain { O, H, W, X, V, Z, L, Q } .

The relation discussed in Illustration . is again a function with domain { a, b, . . .

, z } and the co-domain { , , , . . . , } .

In Illustration . , we discussed three relations, namely (i) y = x (ii) y = x (iii) y = x . Clearly (i) and (ii) are functions whereas (iii) is not a function, if the domain and the co-domain are R . In (iii) for the same x , we have two y values which contradict the definition of the function.

But if we split into two relations, that is, y = √ x and y = − √ x then both become functions with same domain non-negative real numbers and the co-domains [ , ∞ ) and ( −∞ , ] respectively. . . Ways of Representing Functions (a) Tabular Representation of a Function When the elements of the domain are listed like x , x , x .

. . x n , we can use this tabular form.Here, the values of the arguments x , x , x . .

. x n and the corresponding values of the function y , y , y . . .

y n are written out in a definite order. x x . . .

x n y y . . . y n (b) Graphical Representation of a Function When the domain and the co-domain are subsets of R , many functions can be represented using a graph with x -axis representing the domain and y -axis representing the co-domain in the ( x, y ) -plane.

. Functions We note that the first and second figures in Illustration . represent the functions f ( x ) = x and f ( x ) = x respectively. Usually the variable x is treated as independent variable and y as a dependent variable.

The variable x is called the argument and f ( x ) is called the value. (c) Analytical Representation of a Function If the functional relation y = f ( x ) is such that f denotes an analytical expression, we say that the function y of x is represented or defined analytically. Some examples of analytical expressions are x + , sin x + cos x x + , log x + √ x. That is, a series of symbols denoting certain mathematical operations that are performed in a definite sequence on numbers, letters which designate constants or variable quantities.

Examples of functions defined analytically are ( i ) y = x − x + ( ii ) y = − x ( i ii ) y = sin x + cos x ( iv ) A = πr . One of the usages of writing functions analytically is finding domains naturally. That is, the set of values of x for which the analytical expressions on the right-hand side has a definite value is the natural domain of definition of a function represented analytically. Thus, the natural domain of the function, (i) y = x + is ( −∞ , ∞ ) (ii) y = x − is ( −∞ , ∞ ) (iii) y = x − x + is