Is arctan(tan(2x)) an example of a composite function? In other words, would the range of tan2x become the domain of arctanx? Also, is the domain for this composite function the union between the domains of tan2x and arctanx?
not quite. what if $\displaystyle x = \frac{\pi}{4}$...?
also, the range of arctan is (-1,1). but we can easily imagine an original input of, oh say 2.
see for yourself: y = arctan(tan(2x)) - Wolfram|Alpha
the function is not defined for all real x, and is not even equal to 2x except on (-pi/4,pi/4).