quick question

[kaanonball]

how do i find the domain and range of cos(sin^-1 x)

[TrueTears]

the way i do these kind of questions is to split them into a composite function

so let

and

therefore

but we know for f o g to exist ran g must be a subset of dom f

so at the moment

and dom f = R

so yes ran g is a subset of dom f.

and we also know that dom f o g = dom g , hence the domain of the composite function is just the domain of which is [-1,1]

[kaanonball]

ty trutears, but what if the domain of Cos(x) is restricted to [0,pi] and its range restricted to [-1,1]

how can you go about finding the domain and range of the composite function of cos(sin^-1 x) then?

[TrueTears]

ok so we have the same set up as before...

then we still find ran g which is still

but now dom f is

which means now we have to RESTRICT ran g to be a subset of dom f. So ran g must be at least restricted to

now it is a subset of dom f, however to restrict the ran of g we must restrict the domain of g.

so domain of g must be restricted to [0,1]. (note you can also think of it as and just solve for x)

so now we know dom f o g = dom g which is [0,1]

to work out the range is easy, just sub in [0,1] into the composite function and those are your endpoints for the range.

so f o g(0) = 1 and f o g(1) = 0

so range is [0,1]

[kaanonball]

ty =]

[Mao]

Another way to think about it:

construct a right-angled triangle, with an angle and sides


which is the equation for the upper semicircle of the unit circle. Domain [-1,1], range [0,1]

[kaanonball]

thanks Mao, that makes it heaps easier =]

[Flaming_Arrow]

Another way to think about it:

construct a right-angled triangle, with an angle and sides


which is the equation for the upper semicircle of the unit circle. Domain [-1,1], range [0,1]

what if its something like , how would you approach that problem using that method?

[TrueTears]

So the adjacent side is of length x and the hypotenuse is 1.

The other side is

So

[Flaming_Arrow]

So the adjacent side is of length x and the hypotenuse is 1.

The other side is

So

yes but that doesn't give the correct domain/range

[Mao]

So the adjacent side is of length x and the hypotenuse is 1.

The other side is

So

yes but that doesn't give the correct domain/range

because TT found tan(cos^-1x) as opposed to what you asked.


I can't think of how to algebraically find right now, might have a crack at it tomorrow

Suspicion confirmed, there is no simple expression for arctan(sin(x)) or arctan(cos(x)). (fourier expansion yields infinite number of trignometric terms)