Question

Let $f :\{1,2,3,4\} \rightarrow\{1,16,9,4\}$ and $g :\{1,4,9,16\} \rightarrow\left\{\frac{1}{4}, \frac{1}{2}, \frac{1}{3}, 1\right\}$ 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 $\Rightarrow f(1)=16, f(2)=9 \Rightarrow f^{-1}(9)=2$
$\therefore f ^{-1}\left( g ^{-1}\left(\frac{1}{2}\right)\right)= f ^{-1}(9)=2$