Q. 184.9( 9 Votes )

Let f : R – R be a function defined as. Show that f : R – range(f) is one-one and onto. Hence, find f-1.[CBSE 2017]

Answer :

We have f : R – R and

We need to prove f : R – range(f) is invertible.

First, we will prove that f is one-one.

Let x1, x2ϵ A (domain) such that f(x1) = f(x2)

(4x1)(3x2 + 4) = (3x1 + 4)(4x2)

12x1x2 + 16x1 = 12x1x2 + 16x2

16x1 = 16x2

x1 = x2

So, we have f(x1) = f(x2) x1 = x2.

Thus, function f is one-one.

Now, we will prove that f is onto.

Let y ϵ range(f) (co-domain) such that f(x) = y

4x = 3xy + 4y

4x – 3xy = 4y

x(4 – 3y) = 4y

Clearly, for every y ϵ range(f), there exists x ϵ A (domain) such that f(x) = y and hence, function f is onto.

Thus, the function f has an inverse.

We have f(x) = y x = f-1(y)

But, we found f(x) = y


Thus, f(x) is invertible and

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
Different kind of mappings58 mins
Battle of Graphs | various functions & their Graphs48 mins
Functions - 0648 mins
Functions - 1156 mins
Quick Revision of Types of Relations59 mins
Range of Functions58 mins
Functions - 0947 mins
Quick Recap lecture of important graphs & functions58 mins
Range of Quadratic/quadratic & linear/Linear functions45 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