Verified Solution • 2025–26 Session moderate

QJEEM23Ma0971: JEE Main 2023 Mathematics - Class 11 JEEM23Ma0971 Solution: For some a, b, c ∈ N , let f ( x ) = a x − 3 and g ( x ) = x b…

For some a, b, c ∈N, let f(x)=ax−3 and g(x)=xb+c,x∈R. If (fog)−1(x)=(x−72)1/3, then (fog)(ac)+(gof)(b) is equal to ____________.

Step-by-Step Verified Solution

f(x)=ax−3f(x)=a x-3

g(x)=xb+cg(x)=x^{b}+c

(fog)−1=(x−72)13(fog)^{-1}=\left(\frac{x-7}{2}\right)^{\frac{1}{3}}

(fog)−1(x)=(x+3−caa)1b=(x−72)13(fog)^{-1}(x)=\left(\frac{x+3-c a}{a}\right)^{\frac{1}{b}}=\left(\frac{x-7}{2}\right)^{\frac{1}{3}}

⇒a=2,b=3,c=5\Rightarrow a=2, b=3, c=5

fog(ac)+gof(b)fog(a c)+gof(b)

∵f(x)=2x−3\because f(x)=2 x-3

g(x)=x3+5g(x)=x^{3}+5

fog(10)+gof(3)fog(10)+g o f(3)

=2007+32=2007+32

=2039=2039

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