Q. 154.8( 13 Votes )

Consider the func

Answer :

We have f : R+ [–9, ∞) and f(x) = 5x2 + 6x – 9.


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)


5x12 + 6x1 – 9 = 5x22 + 6x2 – 9


5x12 + 6x1 = 5x22 + 6x2


5x12 – 5x22 + 6x1 – 6x2 = 0


5(x12 – x22) + 6(x1 – x2) = 0


5(x1 – x2)(x1 + x2) + 6(x1 – x2) = 0


(x1 – x2)[5(x1 + x2) + 6] = 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 [–9, ∞) (co-domain) such that f(x) = y


5x2 + 6x – 9 = y





Adding to both sides, we get









Clearly, for every y ϵ [–9, ∞), 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,

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 :

Fill in theMathematics - Exemplar

Let f : [2, ∞) <sMathematics - Exemplar

Let f : N Mathematics - Exemplar

Fill in theMathematics - Exemplar

Let f :R →<Mathematics - Exemplar

Let f : [0, 1] <sMathematics - Exemplar

Which of the follMathematics - Exemplar