Question

Which of the following pair of functions has the same graph?
[Note: $[y]$ and $\{y\}$ denotes the greatest integer less than or equal to $y$ and fractional part of y respectively. and sgn denotes the signum function.

• A. $ f(x)=\operatorname{sgn}\left(|x|+\frac{1}{|x|}\right) ; g(x)=\operatorname{sgn}\left(\frac{x^2+1}{2|x|}\right)$
• B. $f(x)=\frac{(\sin x+\cos x)^2-1}{2 \cos ^2 x} ; g(x)=\frac{2 \sin ^2 x}{1-(\sin x-\cos x)^2}$
• C. $ f(x)=e^{\ln \left(\operatorname{sgn} x^2\right)} ; g(x)=\frac{1}{|\operatorname{sgn} x|}$
• D. $f(x)=\sec ^2 x-\tan ^2 x ; g(x)=\cos [\{x\}]$

Answer 1

Answer: (A), (C)

Correct answer is (a) $ f(x)=\operatorname{sgn}\left(|x|+\frac{1}{|x|}\right) ; g(x)=\operatorname{sgn}\left(\frac{x^2+1}{2|x|}\right)$Correct answer is (c) $ f(x)=e^{\ln \left(\operatorname{sgn} x^2\right)} ; g(x)=\frac{1}{|\operatorname{sgn} x|}$