Question

Sketch the graph for $y={\tan }^{-1}(\tan x)$

Answer 1

Hint:
To sketch the graph of $y={\tan }^{-1}(\tan x)$ . First, we have to calculate the domain and range of tanx and ${\tan }^{-1}$ . From this, we can sketch the graph of $\tan x$ and ${\tan }^{-1}$ . With the help of these, we can draw a graph of $y={\tan }^{-1}(\tan x)$
In a relation R from set A to set B, the set of all first components of order pair belongings to R is called the domain and all the second element of pair is called range.

Complete step by step solution:
We know that the domain of tan. function(tangent function) is the set $\left\{x:x\in R,and.x\ne (2n+1){\displaystyle \frac{\pi }{2}},n\in Z\right\}$
here R represents the real number and $x\ne (2n+1){\displaystyle \frac{\pi }{2}}$ means the domain of tan function does not contain an odd multiple of ${\displaystyle \frac{\pi }{2}}$ . $n\in Z$ Represents that n is an integer. Z is the symbol of an integer.
The range of the tan function in R. Here R represents the real number. If we restrict the domain of a tangent function to $\left({\displaystyle \frac{-\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$ then it is one-one and onto with the range R. So, the tangent function is restricted to any of interval etc, is bijective and range is R. SO, ${\tan }^{-1}$ can be defined as the function whose domain is R and range could be any of interval $\left({\displaystyle \frac{-3\pi }{2}},{\displaystyle \frac{\pi }{2}}\right),\left({\displaystyle \frac{-\pi }{2}},{\displaystyle \frac{\pi }{2}}\right),\left({\displaystyle \frac{\pi }{2}},{\displaystyle \frac{3\pi }{2}}\right)$ and so on. This interval gives different branches of the function ${\tan }^{-1}$ . The branch $\left({\displaystyle \frac{-\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$ is called the principal value branch of the function ${\tan }^{-1}$ therefor ${\tan }^{-1}:R\to \left({\displaystyle \frac{-\pi }{2}},{\displaystyle \frac{\pi }{2}}\right)$


The graph of a tan function $y=\tan x$ is given as:

The graph of $y={\tan }^{-1}x$ is given as

As $y={\tan }^{-1}(\tan x)$ is periodic with period π. therefore to draw this graph we should draw the graph for one interval $\pi$ and repeat for the entire value.
As we know that $y={\tan }^{-1}(\tan x)$ = $\left\{x:{\displaystyle \frac{\pi }{2}}<x<{\displaystyle \frac{\pi }{2}}\right\}$
This has been defined for ${\displaystyle \frac{\pi }{2}}<x<{\displaystyle \frac{\pi }{2}}$ that has a length $\pi$ so its graph could be plotted as

Thus this graph of y where y is not defined for $x\in (2n+1){\displaystyle \frac{\pi }{2}}$ .

Note:
One-one function: A function $f:x\to y$ is defined as a one-one function of the image of distinct elements of X under $f$ are distinct.
Onto function: A function is $f:x\to y$ said to be onto the function of every element of Y is an image of some element of X under f.
Bijective Function: The function which is both one-one and onto is called the bijective function.