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# Types of Functions

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#### Types of Functions - Lesson Summary

One-One function :

A function, f : A â†’ 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 âˆˆ A, f(x1) = f (x2) implies x1 = x2.

(Or) for every x1, x2 âˆˆ A, f(x1) â‰  f (x2) implies x1 â‰  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 â†’ R be a function defined by f(x) = 2x.

Let x1, x2 âˆˆ R such that f(x1) = f(x2).

â‡’ 2x1 = 2x2

â‡’ x1 = x2 ( Dividing both sides by 2)

Hence, f(x1) = f(x2) â‡’ x1 = x2 âˆ€ x1, x2 âˆˆ R

âˆ´ 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 âˆˆ B, âˆƒ x âˆˆ 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}.

â‡’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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