range. domain of composite trig functions

[TrueTears]

1. Find the implied domain and range of y=cos(-sin^-1(x)), where cos has a restricted domain [0,pi]

Let y= cos u, where    0<=u<=pi     

and u = -sin^-1(x)

the composite function, cos(-sin^-1(x)), is made up of cos(x) and -sin^-1(x).

let f = cos(x) and g= -sin^-1(x)

therefore for f o g to exist ran g must be a subset of dom f

we have the range of g as [-pi/2,pi/2] and the domain of f as [0,pi]. So for f o g to exist, range of g must be restricted to [0,pi/2].

So for the domain of the composite function, f o g, is the domain of g. Which is [-1,0] (since the range has been restricted to [0,pi/2]).

But now what i don't is why is the range of the composite function, f o g aka y=cos(-sin^-1(x)), [0,1] ? It is the correct range. But how do u get it? Do you do it algebraically, or graphically?

 If algebraically, u just sub in -1 and then 0 into the equation y=cos(-sin^-1(x)) and u get [0,1]. But how do u know that -1 is the lowest point on the graph and 0 is the max point? What if -1 and 0 are the endpoints of the graph and the max point has a different x value?

If graphically, how do u sketch y=cos(-sin^-1(x))? My book does not mention it, it only teaches how to sketch cos^-1, sin^-1 and tan^-1.

Many thanks.