Introduction to Functions

Definition of Function: Domain, Co-domain and Range

Function can be easily defined with the help of the concept of mapping. Let X and Y be any two non-empty sets. "A function from X to Y is a rule or correspondence that assigns to each element of set X, one and only one element of set Y". Let the correspondence be 'f' then mathematically we write f:X → Y

where

y = f(x), x ε X and y ε Y. We say that 'y' is the image of 'x' under 'f' (or x is the pre image of y).

Two things should always be kept in mind:

(i) A mapping f: X → Y is said to be a function if each element in the set X has its image in set Y. It is possible that a few elements in the set Y are present which are not the images of any element in set X.

(ii) Every element in set X should have one and only one image. That means it is impossible to have more than one image for a specific element in set X. Functions can't be multi-valued (A mapping that is multi-valued is called a relation from X to Y)

• Set 'X' is called the domain of the function 'f'.

• Set 'Y' is called the co-domain of the function 'f'.

• Set of images of different elements of set X is called the range of the function 'f'. It is obvious that range could be a subset of co-domain as we may have few elements in co-domain which are not the images of any element of the set X (of course these elements of co-domain will not be included in the range). Range is also called domain of variation.

The set of values for which a function is defined is called the domain of the function. The range of the function is the set of all images of domain of f. In above example, the set A is the domain of the function f. B is not range but the co-domain of the function. The range is the subset of the co-domain. The domain and the range of a function may be an interval, open, closed, semi-closed or semi-open i.e. the domain may be an interval of any of the following types.

        If x ε [q, p], then {x : q < x < p}

        If x ε (-∞, p], then {x : x < p}

        If x ε (-∞, p), then {x : x < p}

        If x ε ]-∞, p[, then {x : x < p}

        If x ε (p, ∞], then {x : x > p}

        If x ε (-∞, ∞), then {x : x ε R}

        If x ε {p, q}, then {x : x = p or x q}

Domain of function 'f' is normally represented as Domain (f). Range is represented as Range (f). Note that sometimes domain of the function is not explicitly defined. In these cases domain would mean the set of values of 'x' for which f(x) assumes real values. e.g. if y = f(x) then Domain (f) = {x : f(x) is a real number}.

e.g. Let X = {a, b, c}, Y = {x, y, z}. Suppose f(a) = y, f(a) = x, f(b) = y, f(c) = z. Then f is not a function of X into Y since a ε X has more than one f-images in Y.

On the other hand, if we set f(a) = x, f(b) = x and f(c) = x, then f:X →  is a function since each element in X has exactly one f-image in Y.

Consider the following examples:

(i) Let X = R, Y = R and y = f(x) = x².

Then f : X → Y is a function since each element in X has exactly one f-image in Y. The range of f = {f(x) : x ε X} = {x² : x ε R} = [0, ∞).

(ii) Let X = R⁺, Y = R⁺ and y = √x. Then f : X → Y is a function. The range of f is R⁺

(iii) Let X = R, Y = R and y² = x. Her f(x) = +√x i.e. f is not a function of X into Y since each x > 0 has two f-images in Y, and further, each x < 0 has no f-image in Y.

We are primarily interested in functions whose domain and ranges are subsets of real numbers. Such functions are often called Real Valued functions.

e.g. Let the function f be defined by f(x) = 1/√(2x+6).

In this formula we must have 2x + 6 > 0 and therefore x > -3. Therefore, the domain of f is (-3, ∞), the range of f = (0, ∞). Thus we have the function f : (-3, ∞) → (0, ∞) defined by f(x) = 1/√(2x+6).

Let the function f be defined by f(x) = x/((x-1)(x-2)). The formula makes sense for all values of x except x = 1 and x = 2. Therefore, the domain of f is R - {1, 2}
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