Question

How do you find the domain and range of ${\tan ^{ - 1}}\left( x \right)$ ?

Answer 1

Hint : 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. The range of a function is the complete set of all possible resulting values of the dependent variable, after we have substituted the domain.

Complete step-by-step answer :
The domain of the expression is all real numbers except where the expression is undefined means There are no radicals or fractions involved. In this case, there is no real number that makes the expression undefined.
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
$\Rightarrow \,\,\,\tan \left( y \right) = x$
Now, let's define the domain and range of $y = \tan x$.
The domain of the function $y = \tan x$ is $x \in \left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)$.
The range of the function $y = \tan x$ is $y \in \left( { - \infty , + \infty } \right)$
As we know, the function $y = {\tan ^{ - 1}}\left( x \right)$ is a symmetry to the function $y = \tan x$ with respect to the line $y = x$.
Therefore, the domain is all the real number i.e., $x \in \mathbb{R}$
And the range is $y \in \left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)$.

Note : The domain where the x values ranges and the range where the y values ranges. Here the function is of the form of a trigonometric function. We must know about the trigonometry and inverse trigonometry concept to solve this question. We should also know about the table for trigonometry ratios for the standard angles.