Q. 115.0( 1 Vote )

If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.

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

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Functions - 0152 mins
Inverse Trigonometric Functions - 0161 mins
Range of Functions58 mins
Functions - 1047 mins
Functions - 0558 mins
Functions - 0748 mins
Functions - 0253 mins
Functions - 0361 mins
Functions - 0840 mins
Functions - 1252 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses