Q. 105.0( 2 Votes )

If f : R <span la

Answer :

We have f : R R and f(x) = x3 – 3.


Recall that a function is invertible only when it is both one-one and onto.


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


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


x13 – 3 = x23 – 3


x13 = x23


(x1 – x2)(x12 + x1x2 + x22) = 0


x1 – x2 = 0 (as x1, x2ϵ R+)


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 R (co-domain) such that f(x) = y


x3 – 3 = y


x3 = y + 3



Clearly, for every y ϵ R, there exists x ϵ R (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


Hence,


Thus, f(x) is invertible and


Hence, we have




Thus, f-1(24) = 3 and f-1(5) = 2.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
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
caricature
view all courses
RELATED QUESTIONS :

| Let * be a binaMathematics - Board Papers

Find the idMathematics - Board Papers

Let f : A Mathematics - Exemplar

Show that the binMathematics - Board Papers

Determine whetherRD Sharma - Volume 1

Fill in theMathematics - Exemplar