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

Answer 1

Answer: (A), (C)

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