Function of One Rreal Variable

In the study of natural phenomena and the solution of technical and mathematical problems, it is necessary to consider the variation of one quantity as dependent on the variation of another. For example, in studies of motion, the path traversed is regarded as a variable, which varies with time. Here we say that the distance traversed is a function of time. The area of a circle, in terms of its radius R, is ∏R^2. If R takes on various numerical values, the area assumes different numerical values. So the variation of one variable brings about a variation in the other. Hence area of the circle is a function of the radius R.

If to each value of variable x (within a certain range) there corresponds a unique value of another variable y, then we say that y is a function of x, or, in functional notation y=f(x). The variable x is called the independent variable y is called the dependent variable. The relation between the variable x and y is called a functional relation. The letter f in the functional notation y=f(x) indicates that some kind of operation must be performed on the value of x in order to obtain the values of y.

The set of all possible values which is independent variable (here ‘x’) is permitted to take for a given functional dependence to be defined is called the domain of definition or simply the domain of function. E.g. the function y=sin x is defined for all values of x. Therefore its domain of definition is the infinite interval -∞<x<∞.

The function y=1/√x-1 is defined for all x>its domain is (1, ∞).

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