Question

Let $f:\{1,2,3,4\}\to \{1,16,9,4\}$ and $g:\{1,4,9,16\}\to \left\{\frac{1}{4},\frac{1}{2},\frac{1}{3},1\right\}$ be two bijective functions such that if $x_1>x_2$, then $f\left(x_1\right)<f\left(x_2\right)$ and $g\left(x_1\right)>g\left(x_2\right)$, then value of $f^{-1}\left(g^{-1}\left(\frac{1}{2}\right)\right)$ equals

• A. 2
• B. 4
• C. 16
• D. 3

Answer 1

Answer: (A)

$g$ is increasing function $\Rightarrow g(1)=\frac{1}{4},g(4)=\frac{1}{3},g(9)=\frac{1}{2}\Rightarrow g^{-1}\left(\frac{1}{2}\right)=9$ and $f$ is decreasing function $\Rightarrow f(1)=16,f(2)=9\Rightarrow f^{-1}(9)=2$
$\therefore f^{-1}\left(g^{-1}\left(\frac{1}{2}\right)\right)=f^{-1}(9)=2$