Answer :

**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 g(x) = |x|

__Check for Injectivity__:

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

⇒ g(x) = g(y)

⇒ |x| = |y|

⇒ x = y

1 belongs to A then

⇒ g(1) = 1 = g( – 1)

Since, it has many element of A co – domain

Hence, g 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

Since g attain only positive values, for negative – 1 in A (co – domain) there is no pre–image in domain A.

Hence, g is not onto function

Thus, It is not Bijective function

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