A function whose graph is symmetrical about the y-axis is given by

f( x ) is symmetric about y – axis then f  (-x ) = f ( x )

$\begin{array}{l}f\left(-x\right)=\cos \left[\log \left(-x+\sqrt{x^2+1}\right)\right]\\ =\cos \left[\log \left[\frac{\sqrt{x^2+1}+x}{\sqrt{x^2+1}+x}\times \left(\sqrt{x^2+1}-x\right)\right]\right]\\ =\cos \left(\log \left(\frac{x^2+1-x^2}{\sqrt{x^2+1}+x}\right)\right)\\ =\cos \left(\log \left(\frac{1}{\sqrt{x^2+1}+x}\right)\right)\\ =\cos \left[-\log \left(\sqrt{x^2+1}+x\right)\right]\\ =\cos \left(\log \left(\sqrt{x^2+1}+x\right)\right)=f\left(x\right)\end{array}$