Question

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

• A. $f\left(x\right)=sin\left[log\left(x+\sqrt{x^2+1}\right)\right]$
• B. $f\left(x\right)=\frac{se{c}^{4}x+cose{c}^{4}x}{x^3+x^{4}cotx}$
• C. $f\left(x+y\right)=f\left(x\right)+f\left(y\right)\forall x,y\in R$
• D. None of these

Answer 1

Answer: (D)

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, $\frac{se{c}^{4}x+cose{c}^{4}x}{x^3+x^{4}cotx}$ 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 $f\left(-x\right)=-f\left(x\right)\therefore f\left(x\right)$ is odd