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One One and Onto Function (Bijection) – Definition and Examples

Here, you will learn one one and onto function (bijection) with definition and examples.

Let’s begin –

What is Bijection Function (One-One Onto Function) ?

Definition : A function f : A B is a bijection if it is one-one as well as onto.

In other words, a function f : A B is a bijection, if it is

(i) one-one i.e. f(x) = f(y) x = y for all x, y A.

(ii) onto i.e. for all y B, there exist x A such that f(x) = y.

Also Read : Types of Functions in Maths – Domain and Range

Example : Let f : A B be a function represented by the following diagram :

bijection function

Solution : Clearly, f is a bijection since it is both one-one (injective) and onto (surjective).

Example : Prove that the function f : Q Q given by f(x) = 2x – 3 for all x Q is a bijection.

Solution : We observe the following properties of f.

One-One (Injective) : Let x, y be two arbitrary elements in Q. Then,

f(x) = f(y) 2x – 3 = 2y – 3 2x = 2y x = y

Thus, f(x) = f(y) x = y for all x, y Q.

So, f is one-one.

Onto (Surjective) : Let y be an arbitrary element of Q. Then,

f(x) = y 2x – 3 = y x = y+32

Clearly, for all y Q, x = y+32 Q. Thus, for all y Q (co-domain) there exist x Q (domain) given by x = y+32 such that f(x) = f(y+32) = 2(y+32) – 3 = y. That is every element in the co-domain has its pre-image in x.

So, f is onto function.

Hence, f : Q Q is a bijection.

Example : Let f : R R be a function defined as f(x) = 2x3+6x2 + 12x +3cosx – 4sinx; then f is-

Solution : We have f(x) = 2x36x2 + 12x + 3cosx – 4sinx

f'(x) = 6x212x + 12 – 3sinx – 4cosx

6x212x + 12 = 6(x1)2 + 6 = g(x) and 3sinx + 4cosx = h(x)

range of g(x) = [6, )

range of h(x) = [-5, 5]

hence f'(x) always lies in the interval [1, )

f'(x) > 0

Hence f(x) is increasing i.e. one-one

Now x f & x - f - & f(x) is continous.

hence its range is R f is onto so f is bijective.

Note :

(i)  If a line parallel to x-axis cuts the graph of the functions atmost at one point, then the f is one-one.

(ii)  If any line parallel to x-axis cuts the graph of the functions atleast at two points, then f is many-one.

(iii)  If continous functions f(x) is always increasing or decreasing in whole domain, then f(x) is one-one.

(iv)  All linear functions are one-one.

(v)  All trigonometric functions in their domain are many one.

(vi)  All even degree polynomials are many one.

(vii)  Linear by linear is one-one.

(viii)  Quadratic by quadratic with no common factor is many one.

(ix)   A polynomial function of degree even define from R R will always be into.

(x)  A polynomial function of degree odd defined from R R will always be onto

(xi)  Quadratic by quadratic without any common factor define from R R is always an into function.

Thus a function can be of these four types :

(i)  one-one onto (Injective and Surjective)(Also known as Bijective mapping)

(ii)  one-one into (Injective but not surjective)

(iii)  many-one onto (surjective but not injective)

(iv)  many-one into (neither surjective nor injective)

 

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