Question

If $xϵ[-1,1],$ then the range of ${\tan }^{-1}(-x)$ is

• A. $[\frac{3\pi }{4},\frac{7\pi }{4}]$
• B. $[\frac{3\pi }{4},\frac{5\pi }{4}]$
• C. $[\pi ,0]$
• D. $[\frac{-\pi }{4},\frac{\pi }{4}]$

Answer 1

The correct option is D $[\frac{-\pi }{4},\frac{\pi }{4}]$

This can be visualized and solved with the help of graphs. we know the graph of $y={\tan }^{-1}\left(x\right)$ is

Now, ${\tan }^{-1}(-x)$ is simply flipping of this curve with respect to the y - axis.
$\therefore y={\tan }^{-1}(-x)$

Now, all we have to do it to see the y - coordinates of the point $x=-1,$ and $x=1.$ Since the function is strictly decreasing we can say all points in between these two will constitute the range of the function.
$\therefore {\tan }^{-1}(-(-1\left)\right)=\frac{\pi }{4}$
${\tan }^{-1}(-(1\left)\right)=-\frac{\pi }{4}$
$\therefore$ Required range = $[\frac{-\pi }{4},\frac{\pi }{4}]$