Question

Find the range of \[{\tan ^{ - 1}}x\]
a) \[( - \pi ,\pi )\]
b) \[R\]
c) \[(0,\pi )\]
d) \[\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)\]

Answer 1

Hint: The range of a function is the complete set of all possible resulting values of the dependent variable, after we have substituted the domain. Also the domain of a function is the complete step of possible values of the independent variable. That is the domain is the set of all possible ‘x’ values which will make the function ‘work’ and will give the output of ‘y’ as a real number.

Complete step by step solution:
Given, \[y = {\tan ^{ - 1}}\left( x \right)\]
Multiply the tan function both side, then
\[ \Rightarrow \tan \left( y \right) = \tan \left( {{{\tan }^{ - 1}}\left( x \right)} \right)\]
On simplification, we get
\[ \Rightarrow \tan \left( y \right) = x\]
Here x denotes the domain of a function.
Let us consider the table for the tangent trigonometry ratio. It is given as

Angle	\[{0^ \circ }\]	\[{30^ \circ }\]	\[{45^ \circ }\]	\[{60^ \circ }\]	\[{90^ \circ }\]
tan	0	\[\dfrac{1}{{\sqrt 3 }}\]	\[1\]	\[\sqrt 3 \]	\[\infty \]

When we see the table the tangent trigonometry ratio can take the maximum value as infinity.
Therefore the value of x will be
\[ \Rightarrow ( - \infty ,\infty )\], this is the maximum and minimum value x can take.
On considering the maximum and minimum value of x we determine the value of y and that will be the range.
When \[x = - \infty \], then
\[ \Rightarrow y = {\tan ^{ - 1}}( - \infty )\]
As we know that \[\tan ( - x) = - \tan x\]. The above term is written as
\[ \Rightarrow y = - {\tan ^{ - 1}}(\infty )\]
By considering the table of trigonometric ratios the above inequality is written as
\[ \Rightarrow y = - \dfrac{\pi }{2}\]
 When \[x = \infty \], then
\[ \Rightarrow y = {\tan ^{ - 1}}(\infty )\]
By considering the table of trigonometric ratios the above inequality is written as
\[ \Rightarrow y = \dfrac{\pi }{2}\]
The y value represents the range of a function.
Therefore the range of \[{\tan ^{ - 1}}x\] is \[\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)\]
Hence the option d) is the correct one.
So, the correct answer is “Option D”.

Note: The domain where the x values range from maximum to minimum and the range where the y values range from maximum to minimum. Here the function is in the form of tangent trigonometry ratio. We must know about the table of trigonometry ratio to solve this question.