Question

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

Answer 1

Answer: 2039

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