Question

Sketch the graph for $y={\tan }^{-1}\left(\tan x\right)$

Answer 1

As $y={\tan }^{-1}\left(\tan x\right)$ is periodic with period $\pi$
$\therefore$ to draw this graph we should draw the graph for one interval $\pi$ and repeat for entire values of $x$.
As we know; $y={\tan }^{-1}\left(\tan x\right)=\{x:-\frac{\pi }{2}<x<\frac{\pi }{2}\}$
Thus, it has been defined for $-\frac{\pi }{2}<x<\frac{\pi }{2}$ that has length $\pi$.So its, graph could be plotted as:
Thus, the curve for $y={\tan }^{-1}\left(\tan x\right)$, where $y$ is not defined for $x\in (2n+1)\frac{\pi }{2}$