Question

If $f:[-1,5]\to [-10,2]$ is an invertible function and $g\left(x\right)$ is inverse function of $f\left(x\right)$, then range of ${\log }_{1/2}\left(\right(g\left(x\right){)}^2+2\left(g\right(x\left)\right)+3)$ is

• A. $[{\log }_{1/2}83,{\log }_{1/2}11]$
• B. $[{\log }_{1/2}38,-1]$
• C. $[{\log }_{1/2}38,1]$
• D. $[-2,1]$

Answer 1

The correct option is B $[{\log }_{1/2}38,-1]$

Range of $g\left(x\right)=[-1,5]$ (since it is inverse of $f\left(x\right)$)
$\left(g\right(x){)}^2+2g(x)+3=(g\left(x\right)+1{)}^2+2$
$\Rightarrow$ Range of $\left(g\right(x){)}^2+2g(x)+3=[2,38]$
$\Rightarrow$ Range of ${\log }_{1/2}\left(g\right(x){)}^2+2g(x)+5)$ is
$[{\log }_{1/2}38,{\log }_{1/2}2]=[{\log }_{1/2}38,-1]$

Hence, option B.