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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: Yoda on February 17, 2014, 07:15:12 pm

Title: domain and range of inverse functions help?
Post by: Yoda on February 17, 2014, 07:15:12 pm

Given that the domain of sinx and cosx are restricted to [-pi/2,pi/2] and [0,pi] respectively, define the implied domain and range of
tan^-1(cosx) 

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Title: Re: domain and range of inverse functions help?
Post by: VCE_2012 on February 18, 2014, 03:08:25 am

Assuming tan^-1(...) is inverse tan ie (arctan(x))
Let f(x)=arctan(cosx)
As we can see this is a "composite function"
By definition the x values are such that the range of cosx is a subset of the domain of arctan(...)
When you perform a simple sketch of arctan(x) you realise the domain consists of all real numbers
So the domain of f(x) is the domain of cosx which is given

Now to the range:
We know that the curve is continuous from zero to pi (domain)

Do a rough sketch of cos (x) on the given domain
Notice that the range  is 1 to -1 and these are the values that are imputed to arctan(...)
From 1 to 0, arctan is decreasing with arctan(1)= pi/4 to arctan(0)=zero
From 0 to -1, arctan is increasing in the negative sense from  zero to arctan(-1)=- pi/4
So the range=[-pi/4,pi/4]