Hyperbola Question 2
Question 2 - 25 January - Shift 1
For some $a, b, c \in \mathbb{N}$, let $f(x)=a x-3$ and $g(x)=x^{b}+c, x \in \mathbb{R}$. If $(f \circ g)^{-1}(x)=(\frac{x-7}{2})^{1 / 3}$ then $(fog)(ac)+(gof)(b)$ is equal to __________
Show Answer
Answer: 2039
Solution:
Formula: Composition of functions
Let $fog(x)=h(x)$
$\Rightarrow h^{-1}(x)=(\frac{x-7}{2})^{\frac{1}{3}}$
$\Rightarrow h(x)=fog(x)=2 x^{3}+7$
$f \circ g(x)=a(x^{b}+c)-3$
$\Rightarrow a=2, b=3, c=5$
$\Rightarrow f \circ g(a c)=f \circ g(10)=2007$
$g(f(x)=(2 x-3)^{3}+5.$
$\Rightarrow gof(b)=gof(3)=32$
$\therefore (fog)(ac)+(gof)(b) = 2039$
