Question

Which of the following functions have the same graph?

• A. $f\left(x\right)={\log }_e{e}^x$
• B. $g\left(x\right)=|x|sgnx$
• C. $h\left(x\right)={\cot }^{-1}\left(\cot x\right)$
• D. $k\left(x\right)=\underset{n\to \infty }{lim}\frac{2|x|}{\pi }\cdot {\tan }^{-1}\left(nx\right)$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $f\left(x\right)={\log }_e{e}^x$
B $g\left(x\right)=|x|sgnx$
D $k\left(x\right)=\underset{n\to \infty }{lim}\frac{2|x|}{\pi }\cdot {\tan }^{-1}\left(nx\right)$

(a) $f\left(x\right)={\log }_e{e}^x=x\cdot {\log }_{e}e=x$
(b) $g\left(x\right)=|x|sgnx$
$=\{\begin{array}{cc}|x|\cdot \frac{x}{|x|},& x\ne 0\\ 0,& x=0\end{array}=\{\begin{array}{cc}x,& x\ne 0\\ 0,& x=0\end{array}$
$\therefore g\left(x\right)=x$,
Which is same as $f\left(x\right)$.
(c) $h\left(x\right)={\cot }^{-1}\left(\cot x\right)$ equal to $x$ only when $x\in (0,\pi )$,
Which is not same as $f\left(x\right)$ and $g\left(x\right)$.
(d) $k\left(x\right)=\underset{n\to \infty }{lim}\frac{2|x|}{\pi }\cdot ta{n}^{-1}\left(nx\right)$
$=\{\begin{array}{cc}\frac{2x}{\pi }\cdot \frac{\pi }{2}& x>0\\ -\frac{2x}{\pi }\cdot -\frac{\pi }{2},& x<0\\ 0,& x=0\end{array}$
$=\{\begin{array}{cc}x,& x>0\\ x,& x<0\\ 0,& x=0\end{array}$
$\therefore k\left(x\right)=x$ for all $x$.
Hence, (a), (b) and (d) are the correct answers.