Summary

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References

**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 x

(Or) for every x_{1}, x_{2} âˆˆ A, f(x_{1}) â‰ f (x_{2}) implies x_{1} â‰ x_{2}.

A one-one function is also known as an injective function or simply an injection.

Ex:

The images of different elements of A under f_{1} and f_{2} are different.

f_{1} and f_{2} 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 *x _{1}, x_{2}* âˆˆ R such that f(x

â‡’ 2x

â‡’ x

Hence, f(x

âˆ´ Function f is one-one.

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.

**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 x

(Or) for every x_{1}, x_{2} âˆˆ A, f(x_{1}) â‰ f (x_{2}) implies x_{1} â‰ x_{2}.

A one-one function is also known as an injective function or simply an injection.

Ex:

The images of different elements of A under f_{1} and f_{2} are different.

f_{1} and f_{2} 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 *x _{1}, x_{2}* âˆˆ R such that f(x

â‡’ 2x

â‡’ x

Hence, f(x

âˆ´ Function f is one-one.

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.