Mathematics - Functions Question with Solution | TestHub

$f\left(x\right)={\left(\log \left(\mathit{sec}x+\mathit{tan}x\right)\right)}^3 \forall x \in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
$f\left(-x\right)= -f\left(x\right),$hence$f\left(x\right)$is odd function
Let$g\left(x\right)=\sec x+\tan x \forall x \in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
$\Rightarrow$$g^{\prime }\left(x\right)=\sec x\left(\sec x+\tan x\right)>0 \forall x \in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
$\Rightarrow$$g\left(x\right)$is one - one function
Hence${\left({\log }_e\left(g\left(x\right)\right)\right)}^3$is one - one function.
And$g\left(x\right) \in \left(0, \infty \right) \forall x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
$\Rightarrow$$\log \left(g\left(x\right)\right) \in R$Hence$f\left(x\right)$is an onto function.