Q. 115.0( 1 Vote )

If f : R →<

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, f : R R, defined by f(x) = 4x3 + 7

To Prove : – f : R R is bijective defined by f(x) = 4x3 + 7

Check for Injectivity:

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

f(x) = f(y)

4x3 + 7 = 4y3 + 7

x3 = y3

x = y

Hence, f is One – One function

Check for Surjectivity:

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

f(x) = y

4x3 + 7 = y

4x3 + 7 – y = 0

Now, we know that for 3 degree equation has a real root

So, let be that root

Thus for clearly , there exist such that f(x) = y

Therefore f is onto

Thus, It is Bijective function

Hence Proved

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