Q. 255.0( 1 Vote )

Consider f: <span

Answer :

Let x, y ϵ N; f(x) = f(y)


9x2 + 6x − 5 = 9y2 + 6y − 5


9(x2 − y2) + 6(x − y) = 0


(x − y) (9x + 9y + 6) = 0


x − y = 0 or x = y


This means the function is one-one, and we can see from the range, it is also onto.


Hence, f(x) is invertible and f.f−1(x) = x


9[f−1(x)]2 + 6[f−1(x)] − 5 = x






OR


Given: ‘*’ is a binary operation on A = Q − {1} defined by


a*b = a − b + ab


Commutativity:


Let any a, b ϵ A,


a*b = a − b + ab b*a = b − a + ab


a − b + ab ≠ b − a + ab


a*b ≠ b*a


‘*’ is not Commutative on A


Associativity:


Let any a, b, c ϵ A


(a*b)*c = (a − b + ab)*c


= (a − b + ab) − c + (a − b + ab)c


= a − b − c + ab + ac − bc + abc


a*(b*c) = a*(b − c + bc)


= a − (b − c + bc) + a(b − c + bc)


= a − b + c + ab − bc − ac + abc


(a*b)*c ≠ a*(b*c)


‘*’ is not Associative on A.


Let e be the identity element in A.


a*e = a = e*a


a − e + ae = e − a + ae


(a − 1)e = 0


e = 0 is the identity element in A.


Let b be the inverse of a


a*b = e = b*a


a − b + ab = 0


b = a ÷ (1 − a)


Every element of A is invertible.


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