Implied Domain and RAnge of Composite Functions

[VCEMan94]

Could someone please explain to me how would i would go about finding the implied domain and range of composite functions
e.g. cos^-1 (sin 2x)
Is there a rule or method to do it? :/

[BubbleWrapMan]

The domain of the cos^-1 function is known to be [-1,1]
The range of the sin function is [-1,1] regardless of whether it's sin(x) or sin(2x)
The domain of the sin function is R
The range of the cos^-1 function is [0,pi]

For composite functions to be defined, the range of the "inside" function has to be a subset or equal to the domain of the "outside" function. So, the range of sin(2x) has to be a subset or equal to the domain of cos^-1(x) -- which it is. So there is no need in this case to restrict the domain of the inside function so that the composite function is defined, but it's something you have to do.

For the final composite function, the domain is the domain of the inside function, and the range is derived from whatever the domain is.

Hope that made sense.