Question

Let $g\left(x\right)=\frac{1}{2}x+7$ and $g^{-1}\left(x\right)=ax-b$. Then number of solutions of $\sin x=\frac{a}{b}$ in the interval $[-100\pi ,100\pi ]$ is

Answer 1

Let $y=\frac{1}{2}x+7$
$\Rightarrow y-7=\frac{1}{2}x$
$\Rightarrow x=2(y-7)$
$\Rightarrow x=2y-14$
replace $x$ with $y$
$\Rightarrow y=2x-14$
$\Rightarrow g^{-1}\left(x\right)=2x-14$
Comparing it with $ax-b$ we have $a=2;b=14$
$\Rightarrow \sin x=\frac{2}{14}=\frac{1}{7}$
$\Rightarrow 2$ solutions are possible for above equation in $x\in [0,2\pi ]$
$\therefore$ Number of solution in $[-100\pi ,100\pi ]=200$