Definition of Functions in Calculus

The concept of function is very important part of Calculus because of its close relation with various phenomena of reality. Before going through the formal definition of function, it is better to understand it through some illustrated examples.

Suppose that a particle is moving at the uniform rate 3 kilometers per hour. If S denotes the distance in kilometers and T time in hours, then S = distance covered in T hours = 3T

When T = 0 hours, S = 0 kilometers
When T = 1 hour, S = 3 kilometers
When T = 2 hours, S = 6 kilometers
When T = 1/3 hours, S= 1 kilometers.

Thus, it is clear that when T changes, S also changes and corresponding to each non negative value of T, there will be only one value of S. In this situation we say that S is a function of T. Here, T is an independent variable over the set of non negative real numbers because T can be any arbitrary non negative real number. S is a dependent variable as its value depends on the value of T. Here S also varies over the set of non negative real numbers. We will call the set over which independent variable T varies the domain and the set over which S varies the range of the function. Here domain and range of the function are both a set of non negative real numbers.

Now we coming to the formal definition of function – Let A and B be two non-empty sets, then a rule ‘f’ which associates with the each element of A with a unique element of B is called a mapping or function from A into B. If f is a mapping from A into B we write f: A → B (read as f is a mapping from A into B). If f associates x Є A to y Є B, then we say that y is the image of the element x under the map f and denote it by f(x) and we write y = f(x). The element x is called as pre-image or inverse-image of y.

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