Question:
Mark the correct alternative in the following question:
Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be given by $f(x)=\tan x .$ Then, $f^{-1}(1)$ is
(a) $\frac{\pi}{4}$
(b) $\left\{n \pi+\frac{\pi}{4}: n \in \mathbf{Z}\right\}$
(c) does not exist
(d) none of these
Solution:
We have,
$f: \mathbf{R} \rightarrow \mathbf{R}$ is given by
$f(x)=\tan x$
$\Rightarrow f^{-1}(x)=\tan ^{-1} x$
$\therefore f^{-1}(1)=\tan ^{-1} 1=\left\{n \pi+\frac{\pi}{4}: n \in \mathbf{Z}\right\}$
Hence, the correct alternative is option (b).
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