Notes On Types of Functions - CBSE Class 12 Maths
One-One function : A function, f : A → B, is defined to be one-one, if the images of the distinct elements of A under f are distinct, i.e. for every x1, x2 ∈ A, f(x1) = f (x2) implies x1 = x2. (Or) for every x1, x2 ∈ A, f(x1) ≠ f (x2) implies x1 ≠ x2. A one-one function is also known as an injective function or simply an injection. Ex: The images of different elements of A under f1 and f2 are different. f1 and f2 are one-one functions. If a function is not one-one, then it is known as many-one. The function f is many-one. Ex: f = {(12 , 2), (15 , 4), (19 , -4), (25 , 6), (9 , 0)} g = {(-1 , 2), (0 , 4), (9 , -4), (18 , 6), (23 , -4)} Hence, the function, g, is not one-one. Ex: Let f:R → R be a function defined by f(x) = 2x. Let x1, x2 ∈ R such that f(x1) = f(x2). ⇒ 2x1 = 2x2 ⇒ x1 = x2 ( Dividing both sides by 2) Hence, f(x1) = f(x2) ⇒ x1 = x2 ∀ x1, x2 ∈ R ∴ Function f is one-one. Horizontal line test: To check the injectivity of the function, f (x) =2x. Draw a horizontal line such that this line cuts the graph only at one place. Such types of functions are known as one-one functions. In thiscase where the line cuts the graph of a function at more than one place, the functions are not one-one. ONTO FUNCTIONS A function f from a set A to a set B is said to be onto, if and only if for every element y of B, there is an element x in A such that f(x) = y. For every y ∈ B, ∃ x ∈ A. So range of f = co-domain of f Or f is onto if and only if f(A) = B. Onto functions are also known as surjective functions or simply surjections. Ex: Co-domain of f = {5, 6, 7} and range of f = {5, 6, 7}. ⇒Co-domain of f = Range of f Such functions are known as onto functions. Ex: The functions in which two elements are mapped to the same element are also onto, provided there exists a pre-image of every element in the codomain of f. f is onto. Ex: Here g is not an 'onto' function since the element, 4, in the codomain of 'g' does not have any pre-image in the domain.

#### Summary

One-One function : A function, f : A → B, is defined to be one-one, if the images of the distinct elements of A under f are distinct, i.e. for every x1, x2 ∈ A, f(x1) = f (x2) implies x1 = x2. (Or) for every x1, x2 ∈ A, f(x1) ≠ f (x2) implies x1 ≠ x2. A one-one function is also known as an injective function or simply an injection. Ex: The images of different elements of A under f1 and f2 are different. f1 and f2 are one-one functions. If a function is not one-one, then it is known as many-one. The function f is many-one. Ex: f = {(12 , 2), (15 , 4), (19 , -4), (25 , 6), (9 , 0)} g = {(-1 , 2), (0 , 4), (9 , -4), (18 , 6), (23 , -4)} Hence, the function, g, is not one-one. Ex: Let f:R → R be a function defined by f(x) = 2x. Let x1, x2 ∈ R such that f(x1) = f(x2). ⇒ 2x1 = 2x2 ⇒ x1 = x2 ( Dividing both sides by 2) Hence, f(x1) = f(x2) ⇒ x1 = x2 ∀ x1, x2 ∈ R ∴ Function f is one-one. Horizontal line test: To check the injectivity of the function, f (x) =2x. Draw a horizontal line such that this line cuts the graph only at one place. Such types of functions are known as one-one functions. In thiscase where the line cuts the graph of a function at more than one place, the functions are not one-one. ONTO FUNCTIONS A function f from a set A to a set B is said to be onto, if and only if for every element y of B, there is an element x in A such that f(x) = y. For every y ∈ B, ∃ x ∈ A. So range of f = co-domain of f Or f is onto if and only if f(A) = B. Onto functions are also known as surjective functions or simply surjections. Ex: Co-domain of f = {5, 6, 7} and range of f = {5, 6, 7}. ⇒Co-domain of f = Range of f Such functions are known as onto functions. Ex: The functions in which two elements are mapped to the same element are also onto, provided there exists a pre-image of every element in the codomain of f. f is onto. Ex: Here g is not an 'onto' function since the element, 4, in the codomain of 'g' does not have any pre-image in the domain.

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