What is a mathematical function? A function is a technical term used to define relation between variables. Let us more understand what is a Math Function? A variable y is called a function of a variable if for every value of x there is a definite value of y. example of mathematical functions y = x^2.x is called the independent variable as it takes any arbitrary assigned value whereas y is called the dependent variable as its value depends upon the value of x. The set of all possible value of the independent variable in a function is called the domain of the function and the set of values of the dependent variable is called the range of the function.
Let us more understand about functions math. We can classify functions in math as follow:
1. Into functions
A function f : X -> Y is called an into function if there is atleast one element in Y which has no pre-image in X. A function is an into function if its range is a proper subset of codomain.
2. Onto functions(Surjection)
A function f : X -> Y is called an onto function if every element in Y has atleast one pre-image in X i.e. every element of Y is image of some element of X under f. i.e. for every y belongs to Y, there exists an element x in X such that
f(x) = y.
3. One-one function (Injection)
A function f : X -> Y is called a one-one function if no element of X has more than one image in Y. In other words, if the images of distinct elements in X under f are distinct i.e. for every x1, x2 belongs to X.
4. Many-one function
A function f : X -> Y is called a many-one function if two or more elements of X have the same image in Y.
5. One-one into function
When a function is both 1-1 and into function, then it is called one-one into function. It satisfies the following properties:
(i) No two elements of the domain have the same image.
(ii) There is atleast one element in codomain which is not the image of any element of the domain.
6. One-one onto function
It is a function which is both 1-1 and onto. It satisfies the following properties:
(i) No two elements of the domain have the same image.
(ii) It is one-one i.e. f(x) = f(y)
(iii) It is onto i.e. for all y belongs to Y, there exists x belongs to X such that f(x) = y.
7. Many one into function
It is a function which is both many-one and into. It has the following properties:
(i) There are atleast two elements of the domain which correspond to the same element of the codomain.
(ii) There is atleast one element of the codomain which is not the image of any element of the domain.
8. Many-one onto function
It is a function which is both many-one and onto. It satisfies the following properties:
(i) There are atleast two elements of the domain which correspond to the same element of the codomain.
(ii) Every element of the codomain corresponds to some element of the domain.
9. Identity function
A function f : X -> X such that f(x) = x for all x belongs to X is called the identity function. In an identity function each element of the domain corresponds to itself.
10. Constant function
A function f in which all elements of X are associated with the same element of Y is called a constant function.
11. Equal function
Two functions f and g are said to be equal if their domains are same and f(x) = g(x) for all x. If f and g are equal we write them as f = g.
Know more about Continuity definition and function in calculus. If you have problem in solving calculus problems leave a comment with the problem. I will try to solve.
Let us more understand about functions math. We can classify functions in math as follow:
1. Into functions
A function f : X -> Y is called an into function if there is atleast one element in Y which has no pre-image in X. A function is an into function if its range is a proper subset of codomain.
2. Onto functions(Surjection)
A function f : X -> Y is called an onto function if every element in Y has atleast one pre-image in X i.e. every element of Y is image of some element of X under f. i.e. for every y belongs to Y, there exists an element x in X such that
f(x) = y.
3. One-one function (Injection)
A function f : X -> Y is called a one-one function if no element of X has more than one image in Y. In other words, if the images of distinct elements in X under f are distinct i.e. for every x1, x2 belongs to X.
4. Many-one function
A function f : X -> Y is called a many-one function if two or more elements of X have the same image in Y.
5. One-one into function
When a function is both 1-1 and into function, then it is called one-one into function. It satisfies the following properties:
(i) No two elements of the domain have the same image.
(ii) There is atleast one element in codomain which is not the image of any element of the domain.
6. One-one onto function
It is a function which is both 1-1 and onto. It satisfies the following properties:
(i) No two elements of the domain have the same image.
(ii) It is one-one i.e. f(x) = f(y)
(iii) It is onto i.e. for all y belongs to Y, there exists x belongs to X such that f(x) = y.
7. Many one into function
It is a function which is both many-one and into. It has the following properties:
(i) There are atleast two elements of the domain which correspond to the same element of the codomain.
(ii) There is atleast one element of the codomain which is not the image of any element of the domain.
8. Many-one onto function
It is a function which is both many-one and onto. It satisfies the following properties:
(i) There are atleast two elements of the domain which correspond to the same element of the codomain.
(ii) Every element of the codomain corresponds to some element of the domain.
9. Identity function
A function f : X -> X such that f(x) = x for all x belongs to X is called the identity function. In an identity function each element of the domain corresponds to itself.
10. Constant function
A function f in which all elements of X are associated with the same element of Y is called a constant function.
11. Equal function
Two functions f and g are said to be equal if their domains are same and f(x) = g(x) for all x. If f and g are equal we write them as f = g.
Know more about Continuity definition and function in calculus. If you have problem in solving calculus problems leave a comment with the problem. I will try to solve.
