# Let $\; f:R \rightarrow R$ be given as $\;f(x)=\tan x$. Then $\;f^{-1}(1)$ is

\begin{array}{1 1}(A)\;\frac{\pi}{4} & (B)\;{n\pi+\frac{\pi}{4}:n\in Z}\\(C)\;does\;not\;exist & (D)\;none\; of\; these\end{array}

Toolbox:
• $f(x)=\tan x$
• To find $f^{-1}(1)$ we check if for what values
• $(fof^{-1})(1)=1=>f(f^{-1}(1))=1$
• $or =\tan (f^{-1}(1))=1$
$f:R \to R$

$f(x)=\tan x$

since $f^{-1}$ is inverse of f

$(fof^{-1})(1)=1$

$f(f^{-1}(1))=1$

$\tan (f^{-1}(1))=1$

But $\tan(\frac{\pi}{4})=1$

$\tan (f^{-1}(1))=\tan \frac{\pi}{4}$

=>$f^{-1}(1)=n\pi+\frac{\pi}{4} \; n \in z$

'B' option is correct

If $\tan \theta =\tan \alpha$

$\theta= n \pi +\alpha \; n \in z$