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 $g(x)=2 \sin ^2 x+\cos 2 x$
• C. $ f(x)=2^{\operatorname{sgn}\left(x^2+3 x+3\right)} \text { and } g(x)=2 \operatorname{sgn}\left(x^2+3 x+3\right)$
• D. $f(x)=\frac{\operatorname{sgn} x}{\operatorname{sgn} x}$ 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 $f = R -\left\{ x \mid x =(2 n +1) \frac{\pi}{2}, n \pi\right.$ where $\left.n \in I \right\}$ 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.