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, f : R → R, defined by f(x) = sin x
Check for Injectivity:
Let x,y be elements belongs to R i.e such that
⇒ f(x) = f(y)
⇒ sin x = sin y
⇒ x ≠ y
Hence, f is not One – One function
Check for Surjectivity:
Let y be element belongs to R i.e be arbitrary, then
⇒ f(x) = y
⇒ sin x = y
Now, for y>1 x not belongs to R (Domain)
Hence, f is not onto function
Thus, It is also not Bijective function
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