ATAR Notes: Forum
VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: TrueTears on December 25, 2008, 12:11:32 am
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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.