Question

If $xϵ[-1,1],\text{then range of}ta{n}^{-1}(-x)$ 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 $y=ta{n}^{-1}\left(x\right)$ is

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

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.
$\therefore ta{n}^{-1}(-(-1\left)\right)=\frac{\pi }{4}$
$ta{n}^{-1}(-(1\left)\right)=-\frac{\pi }{4}$
$\therefore$ Required range = $[\frac{-\pi }{4},\frac{\pi }{4}]$