Question

For some $a, b, c \in N$, let $f(x)=a x-3$ and $g (x)=x^{ b }+ c , x \in R$. If $(f \circ g)^{-1}(x)=\left(\frac{x-7}{2}\right)^{1 / 3}$, then $(f \circ g)(a c)+(g \circ f)(b)$ is equal to _____

Answer 1

Let $f \circ g(x)=h(x)$
$ \Rightarrow h^{-1}(x)=\left(\frac{x-7}{2}\right)^{\frac{1}{3}} $
$\Rightarrow h(x)=\text { fog }(x)=2 x^3+7$
$ \text { fog }(x)=a\left(x^{ b }+ c \right)-3 $
$ \Rightarrow a =2, b =3, c =5$
$ \Rightarrow \operatorname{fog}( ac )= fog (10)=2007$
$ g \left( f ( x )=(2 x -3)^3+5\right. $
$ \Rightarrow \operatorname{gof}( b )=\operatorname{gof}(3)=32$
$ \Rightarrow \operatorname{sum}=2039$