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Q.
A function whose graph is symmetrical about the y-axis is given by
Relations and Functions - Part 2
Solution:
A function whose graph is symmetrical about the y-axis must be even
Since sin x and $log\left(x+\sqrt{x^{2}+1}\right)$ are odd function
therefore $sin \left(log\left(x+\sqrt{x^{2}+1}\right)\right)$ must be odd.
Also, $\frac{sec^{4} \,x + cos\,ec^{4} x}{x^{3}+x^{4}\,cot\,x}$ is an odd function.
Now, let $f\left(x+y\right) = f\left(x\right) + f\left(y\right)\,\forall x, \,y \in R$
$\therefore f\left(0+0\right) = f\left(0\right) + f\left(0\right) \,\therefore \,f\left(0\right) = 0$
$f\left(x-x\right) = f\left(x\right) + f\left(-x\right)$ or $0 = f\left(x\right)+f\left(-x\right)$
i.e $f\left(-x\right) = -f\left(x\right) \quad\therefore f\left(x\right)$ is odd