Functions in Grade X

Grade X is important grade in your career, in this grade you learn the basic concept of almost every topic that you are going to learn in higher classes of ICSE board syllabus, if you choose math as your subject. Today we are going focus one of such topic that you will study in Grade XI, if you chose math. Continuous/discrete functions are the topic that you are going to learn in this article. Before we move to this topic let’s have a brief introduction of functions. Functions are one of the most important classes of mathematical object, which are extensively used in almost all sub fields of mathematics. By the names of both discrete functions and continuous functions we can say that both are two special types of functions. Also try Increasing and Decreasing Functions

When we define a relation between two sets of functions in such a way that for each element in the first set, the value that corresponds to it in the second set is unique is defined as a function. Suppose ‘f’ be a function defined from the set A into set B. Then for each xϵA, the symbol f(x) denotes the unique value in the set B that corresponds to x. It is called the image of x under f. Therefore, a relation f from A into B is a function, if and only if for, each xϵA and y ϵA; if x = y then f(x) = f(y). The set A is called the domain of the function f, and it is the set in which the function is defined.

Now, what is discrete function? A ‘discrete function’ is a function whose domain is at most countable. By this we mean that it is possible to make a list that includes all the elements of the domain. Generally a finite set is countable and infinite not. For instance: The set of all or some natural numbers and the set of rational numbers are utmost countable infinite sets. on the other side the set of irrational numbers and the set of real numbers are not at most countable. One of the most common discrete functions is the factorial function. f :N U0→N recursively defined by f(n) = nf(n-1) for each n ≥ 1 and f(0)=1 is called the factorial function. Observe that its domain N U0 is at most countable.

Now, the other type of function which is called as continuous function. Let ‘f’ be a function in such a way that for each k in the domain of f, f(y)→f(c) as y → c. Then  ‘f’ is a continuous function. This means that it is possible to make f(y) arbitrarily close to f(c) by making y sufficiently close to c for each c in the domain of f. Consider the function f(y) = y + 2 on R. It can be seen that as y → c, y + 2 → c + 2 that is f(y)→f(c). Therefore, f is a continuous function. Now, consider g on positive real numbers g(y) = 1 if y > 0 and g(y) = 0 if y = 0. Then, this function is not a continuous function as the limit of g(y) does not exist (and hence it is not equal to g(0) ) as y → 0.

This is all about the topic. For any query take help of your teacher and i am also here to help you.

In next post we will discuss about Expressions and Functions in Grade X. Visit our blogs for more information on mean in statistics