Question

Which of the following are identical functions?
(where $[ x$ ] denotes greatest integer less than or equal $x ,\{ x \}$ denotes fractional part of $x$ and $\operatorname{sgn} x$ denotes signum function of $x$ respectively.)

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

Answer 1

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

(A) $ \operatorname{sgn}(|x|+1)=1 \forall x \in R$.
(B) $ \sin ^2(\ln x )+\cos ^2(\ln x )=1 \forall x \in R ^{+}$
(C) $\frac{2}{\pi}\left(\sin ^{-1}\{x\}+\cos ^{-1}\{x\}\right)=1 \forall x \in R$.
(D) $\sec ^2[\{x\}]-\tan ^2\{[x]\}=\sec ^2 0^{\circ}-\tan ^2 0^{\circ}=1 \forall x \in R$.