Q. 8 A3.7( 3 Votes )

# Let A = [–1, 1], Then, discuss whether the following functions from A to itself are one – one, onto or bijective:

TIP: One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.

So, is One – One function

a≠b

f(a)≠f(b) for all

f(a) = f(b)

a = b for all

Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, is Surjection iff for each , there exists such that f(a) = b

Bijection Function: – A function is said to be a bijection function if it is one – one as well as onto function.

Now, here f: A A: A = [–1, 1] given by function is

Check for Injectivity:

Let x, y be elements belongs to A i.e. such that

f(x) = f(y)

2x = 2y

x = y

1 belongs to A then

Not element of A co – domain

Hence, f is not One – One function

Check for Surjectivity:

Let y be element belongs to A i.e be arbitrary, then

f(x) = y

x = 2y

Now,

1 belongs to A

x = 2, which not belong to A co – domain

Hence, f is not onto function

Thus, It is not Bijective function

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