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
⇒ 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
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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