Answer
The domain of $~~tan^{-1}(tan~x)~~$ is all real numbers except $\frac{\pi}{2}+\pi~n,$ where $n$ is an integer.
The range of $~~tan^{-1}~(tan~x)~~$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$
We can see a sketch of the graph of $~~tan^{-1}~(tan~x)~~$ below.
Since the functions in Exercise 77 and Exercise 78 have different domains and ranges, the resulting graphs are not the same.
Work Step by Step
$y = tan^{-1}(tan~x)$
The domain of $~tan~x~$ is all real numbers except $\frac{\pi}{2}+\pi~n,$ where $n$ is an integer.
The range of $tan~x$ is all real numbers.
The domain of $~~tan^{-1}x~~$ is all real numbers
Therefore, the domain of $~~tan^{-1}(tan~x)~~$ is all real numbers except $\frac{\pi}{2}+\pi~n,$ where $n$ is an integer.
The range of $~~tan^{-1}~x~~$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$
Therefore, the range of $~~tan^{-1}~(tan~x)~~$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$
We can see a sketch of the graph of $~~tan^{-1}~(tan~x)~~$ below.
Exercise 77:
$y = tan(tan^{-1}~x)$
The domain of $~~tan^{-1}x~~$ is all real numbers
The range of $~~tan^{-1}~x~~$ is $~~(-\frac{\pi}{2}, \frac{\pi}{2})$
Therefore, the domain of $~~tan(tan^{-1}~x)~~$ is all real numbers.
On the domain $(-\frac{\pi}{2}, \frac{\pi}{2})$, the range of $tan~x$ is all real numbers.
Therefore, the range of $~~tan(tan^{-1}~x)~~$ is all real numbers.
Since the functions in Exercise 77 and Exercise 78 have different domains and ranges, the resulting graphs are not the same.
