Question

Which of the following pair of functions have the same graph?
[Note : [k], $\{k\}$ and sgn $k$ denote the largest integer less than or equal to $k$, fractional part of $k$ and signum function of $k$ respectively.]

• A. $f(x)=\ln [1+\{x\}]\text{ and }g(x)=\ln (1+[\{x\}])$
• B. $f(x)=\frac{1}{1+{\tan }^{2}x}+\frac{1}{1+{\cot }^{2}x}$ and $g(x)=2{\sin }^{2}x+\cos 2x$
• C. $f(x)=2^{\mathrm{sgn}\left(x^2+3x+3\right)}\text{ and }g(x)=2\mathrm{sgn}\left(x^2+3x+3\right)$
• D. $f(x)=\frac{\mathrm{sgn}x}{\mathrm{sgn}x}$ and $g(x)=\frac{|x|}{|x|}$

Answer 1

Answer: (A), (C), (D)

(A) Clearly, domain of $f=$ domain of $g=R$.
and range of $f=$ range of $g=\{0\}$. Also $f(x)=g(x)\forall x\in R$, so $f(x)$ and $g(x)$ are identical functions.
(B) Domain of $f=R-\{x\mid x=(2n+1)\frac{\pi }{2},n\pi$ where $n\in I\}$ and domain of $g=R$.
(C) Clearly, domain of $f=$ domain of $g=R$. and range of $f=$ range of $g=\{2\}$. Also $f(x)=g(x)\forall x\in R$, so $f(x)$ and $g(x)$ are identical functions.
(D) Clearly, domain of $f=$ domain of $g=R-\{0\}$. and range of $f=$ range of $g=\{1\}$. Also $f(x)=g(x)\forall x\in R$, so $f(x)$ and $g(x)$ are identical functions.