Question

Let $f\left(x\right)=\log x$ and $g\left(x\right)=\sqrt{x}$ are two functions such that the composite function $\left(fog\right)\left(x\right)$ exists, then

• A. $\left(fog\right)\left(x\right)$ is monotonically increasing
• B. $\left(fog\right)\left(x\right)$ is monotonically decreasing
• C. $\left(fog\right)\left(x\right)$ is non-monotonic
• D. None of the above

Answer 1

The correct option is A $\left(fog\right)\left(x\right)$ is monotonically increasing

$f\left(x\right)=\log x,x>0$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{x}>0,\forall x$ in it's domain
$g\left(x\right)=\sqrt{x}$
$\Rightarrow g^{\prime }\left(x\right)=\frac{1}{2\sqrt{x}}>0,\forall x$ in it's domain
$\therefore f$ and $g$ are strictly increasing functions
$\Rightarrow \left(fog\right)\left(x\right)$ is also monotonically increasing