Question

Is $f\left(x\right)=\ell n(x+\sqrt{1+x^2})$ odd or even?

Answer 1

$\begin{array}{c}f\left(x\right)=\ln \left(x+\sqrt{1+x^2}\right)oddoreven\\ Putx=-x\\ \Rightarrow f\left(x\right)=\ln \left(-x+\sqrt{1+x^2)}=\ln \left(x+\sqrt{x^2+1}\right)\right)\\ =-\ln \left(x+\sqrt{1+x^2}\right)=-f\left(x\right)\end{array}$

Hence, it is odd function.