Question

If f : $(e,\infty )\to R$ & $f\left(x\right)=\log \left[\log \right(\log x\left)\right]$, then $f$ is

• A. One - one but not onto
• B. Onto but not one-one
• C. One - one & onto both
• D. Neither one - one not onto

Answer 1

The correct option is C One - one & onto both

$f\left(x\right)=\log \left[\log \right(\log x\left)\right]$
$f^{\prime }\left(x\right)=\frac{1}{\log \left(\log x\right)}.\frac{1}{\log x}.\frac{1}{x}$
Domain of f is $(e,\infty )$. Clearly in domain of f, $f^{\prime }\left(x\right)>0$
Thus f is strictly increasing function.$\Rightarrow$ one-one
Also Range of f is $R=$ co-domain of f $\Rightarrow$ onto
Hence f is both one-one and onto.