Function

Relation and function of Class 12

Functions are the major tools to describe the real world in mathematical terms.

The temperature at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). The interest paid on a cash investment depends on the length of time the investment is held. In each case, the value of one variable quantity, which we might call y, depends on the value of another variable quantity, which we might call x. Since the value of y is completely determined by the value of x, we say that y is a function of x. Here y is called dependent variable and x is called independent variable.

Let X and Y be two non-empty sets. A function f from X to Y written as f : X → Y, is a rule or a correspondence which connect every member, say, x of X to exactly one member, say, y of Y. e.g. When we study circles, if we take area as y and the radius as x, we have y = πx2, we say that y is a function of x. The equation y = πx2 is a rule (correspondence) that tells how to calculate a unique (single) output value of y for each possible input value of the radius x. Here we say y is a function of x and represent it as y = f(x), y = g(x) or y = h(x) normally.

The set of all possible input values of x for which f(x) exists or is defined is called the domain of the function. The set of all out put values of the y is the range of the function. Since in case of radii it can not be negative so domain is [0, ∞). And so range is also [0, ∞).

X is domain of the function. f(X) is range of the function and Y is co-domain of the function. Range is always subset of co-domain.

f(x) is also called as image of x under f or the f-image of x. Likewise x is called pre-image
of y or f(x).

Important Points

(i) f : X → Y is a function if each element x in X has a unique image f(x) in Y.

(ii) f : X → Y is not a function if there is an element in X which does not have an f-image in Y.

(iii) f : X → Y is not a function if there is an element in X which has more than one

f-image in Y.

e.g.

(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 ∈ X} = {x2 : x ∈ R} = [0, ∞).

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

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