Question

Let $y=f(x)$ be a function satisfying the equation $\tan ^{-1} y=\tan ^{-1} x+C$ where $y=1$ when $x=0$. Range of the function $f ( x )$, is

• A. $R -\{-1\}$
• B. $(-1, \infty)$
• C. $[1, \infty)$
• D. $(-\infty, 1]$

Answer 1

Answer: (B)

$\tan ^{-1} y =\tan ^{-1} x + C$
$x =0 ; y =1 \Rightarrow C =\frac{\pi}{4} $
$\Rightarrow \tan ^{-1} y =\tan ^{-1} x +\frac{\pi}{4} \Rightarrow
$
note: even $\frac{-\pi}{2}< tan ^{-1} x +\frac{\pi}{4}<\frac{\pi}{2} ; \frac{-\pi}{2}<\tan ^{-1} x <\frac{\pi}{4} ;-\infty< x < 1 \Rightarrow$ (A) $x <1$
Hence $y=\tan \left(\tan ^{-1} x+\frac{\pi}{4}\right)=\left(\frac{x+1}{1-x}\right) \Rightarrow x=\frac{y-1}{y+1}<1 $
$\Rightarrow \frac{2}{y+1}>0 \Rightarrow y \in(-1, \infty)$