## Algebra

FUNCTIONS-1

Function is basically a relation between two sets, set of inputs and a set of output where an element of the set containing the inputs corresponds to not more than one output. Let us say we have a set of four persons namely called A whose elements are P,Q, R and T and set a set B containing names of five companies namely X1,X2,X3,X4 and X5. Now a relation between A to B is a function if a member of A is linked to not more than one member of B.

Here the elements X4 and X5 are not mapped but none of the elements P,Q,R and S are mapped to more than one element in the set B.

This means every algebraic expression may not be a function. For one value of x one must get only one value of y. It can so happen that for more than one value of x, we can get one value of y. This means for an algebraic expression, to be a function, a line drawn parallel to the Y axis must cut the curve at only one point. Consider the expression x^{3}, its curve is as shown below and lines drawn parallel to the Y axis will cut the curve at only one point.

Thus, y = x^{3} is a function.

Consider the expression: y^{2 }= 4ax. Here the lines drawn parallel to the Y axis cut the curve at more than one point

Hence y^{2 }= 4ax is not a function.

Function can be viewed as a machine which takes input, processes it and gives output. The set of input for which the function is defined, is known as Domain while the set of values of output which we get is known as Range.

For e.g. let’s say we have a function y = x^{2} +1 , then the function takes the input as the values of x, squares it and adds 1 to it and gives the output.

As one can see, the input can be any real number so the Domain of the given function is R (Set of all real numbers) while the range is positive number starting from 1.

__Calculation of Domain:__