Class 12 Relations And Functions · Mcq Single Correct1 Marksfunctional_equations

Class 12 Maths · Relations And Functions · Mcq Single Correct #101

Let \(a,b\) be positive integers and \(f:(0,\infty)\to(0,\infty)\) satisfy \(f(xf(y))=x^a y^b\) for all positive \(x,y\). Which identity must hold for every such choice of \(a,b,f\)?

(A) \(b^2=a\)
(B) \(a=b\)
(C) \(a^2=b\)
(D) None of these

✅ Step-by-Step Solution

Final Answer: Option (C): \(a^2=b\)

Write \(c=f(1)>0\). Setting \(y=1\) gives \(f(xc)=x^a\). Taking \(x=1/c\) gives \(c=c^{-a}\), so \(c=1\) because \(a>0\). Hence \(f(x)=x^a\). Substitution gives \(x^a y^{a^2}=x^a y^b\) for all positive \(x,y\), forcing \(b=a^2\).

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Class 12 Maths · Relations And Functions · Mcq Single Correct #101

✅ Step-by-Step Solution

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