Question

How do you find the inverse of arc \[\tan \left( x+3 \right)\] and is it a function?

Answer 1

Hint: Assume the given function as y and convert arc \[\tan \] into \[{{\tan }^{-1}}\] function. Take tangent function both the sides and use the property \[\tan \left( {{\tan }^{-1}}x \right)=x\] for \[x\in \left( -\infty ,\infty \right)\]. Find the value of x in terms of y and write the obtained function equal to \[{{f}^{-1}}\left( y \right)\]. Finally, replace the variable y with x to get the required inverse function. To check if it is a function or not check that for no value of x there are more than one value of the obtained inverse function.

Complete step by step answer:
Here, we have been provided with the expression arc \[\tan \left( x+3 \right)\] and we are asked to find its inverse function and we have to check if this obtained inverse expression will really be a function or not.
Now, we know that arc \[\tan \] represents the inverse trigonometric function \[{{\tan }^{-1}}\], so assuming the given expression as y, we have,
\[\Rightarrow y={{\tan }^{-1}}\left( x+3 \right)\]
Since, the range of \[{{\tan }^{-1}}\] function is \[\left( \dfrac{-\pi }{2},\dfrac{\pi }{2} \right)\], so here \[y\in \left( \dfrac{-\pi }{2},\dfrac{\pi }{2} \right)\]. Taking tangent function both the sides, we have,
\[\Rightarrow \tan y=\tan \left[ {{\tan }^{-1}}\left( x+3 \right) \right]\]
Using the identity: - \[\tan \left( {{\tan }^{-1}}x \right)=x\], we get,
\[\begin{align}
  & \Rightarrow \tan y=x+3 \\
 & \Rightarrow x+3=\tan y \\
\end{align}\]
\[\Rightarrow x=\tan y-3\] - (1)
Now, \[{{\tan }^{-1}}\left( x+3 \right)\] is a function of x, therefore we can write it as \[f\left( x \right)\], so, we have,
\[\begin{align}
  & \Rightarrow f\left( x \right)={{\tan }^{-1}}\left( x+3 \right) \\
 & \Rightarrow f\left( x \right)=y \\
\end{align}\]
\[\Rightarrow x={{f}^{-1}}\left( y \right)\] - (2)
From equations (1) and (2), we get,
\[\Rightarrow {{f}^{-1}}\left( y \right)=\tan y-3\]
Replacing y with x, we get,
\[\Rightarrow {{f}^{-1}}\left( x \right)=\tan x-3\]
Hence, the above relation represents the inverse of the given function.
Now, we know that there is not any value of x for which the expression \[\tan x-3\] will have 2 or more values but there can be many values of x for which \[\tan x-3\] will have the same value. Therefore, \[{{f}^{-1}}\left( x \right)\] is a many – one function.
Hence, we can say that \[{{f}^{-1}}\left( x \right)\] is a function.

Note:
 One may note that here we were not provided with the information regarding the range of the given expression arc \[\tan \left( x+3 \right)\] and that is why we need to consider it ourselves that \[y\in \left( \dfrac{-\pi }{2},\dfrac{\pi }{2} \right)\]. You must remember the definition of a function and its difference in relation to solving the second part of the question, i.e., to determine if it is a function or not. At last, after finding \[{{f}^{-1}}\left( y \right)\] do not forget to replace y with x.