What is the domain of the function composed of
f(x) = sinx
g(x) = \sqrt{1 - x^2}

f(g(x)) dom = ?

Question

What is the domain of the function composed of 
f(x) = sinx
g(x) = \sqrt{1 - x^2}

f(g(x)) dom = ?

anonymous · Answer

make sure that $1-x^2\geq 0$

anonymous · Answer

do you know how to finish that?

anonymous · Answer

Wait
For this one, it will be [-1,1]

anonymous · Answer

yes

anonymous · Answer

Wait, so then g(f(x)) = R right, because for any value of of the function, it will work?
So the trick with the domains of composite functions is that the domain is always determined by the innermost function?

anonymous · Answer

well close. it is everthing in the inner function whose output is in the domain of the outer function

anonymous · Answer

but in this case since for all x $-1\leq \sin(x)\leq 1$ the domain of $g\circ f$ is all real numbers

anonymous · Answer

I don't really understand what you said about the composite functions. Could you give me an example to illustrate it?

anonymous · Answer

well here is an easy one. suppose you have 
$$f(x)=\sqrt{x}$$ and $$g(x)=1-x$$ now the domain of g is clearly all real numbers, and the domain of f is all numbers greater than or equal to zero
but the domain of 
$$f\circ g$$ is not all real numbers even though the domain of g is 
it is the set of all inputs for g are in the domain of f, that is the set where 
$$1-x\geq 0$$ i.e $$1\geq x$$

anonymous · Answer

that was a simple example but i hope you get the idea.  in this case of course 
$$f\circ g(x)=\sqrt{1-x}$$ so the domain is $x\leq 1$

anonymous · Answer

So it basically has to be in the range of the outer function?

anonymous · Answer

That is, the x-values that you sub in for the innermost function?

anonymous · Answer

exactly

anonymous · Answer

OH, so you can just do:
range(outer most fcn) interset (domain(inner most function)

anonymous · Answer

intersect, with that upside down U

anonymous · Answer

Is that true?

anonymous · Answer

well not exactly still. it is what i wrote above
the domian of the composite funciton is all numbers in the domian of the "inner function" that produce ouputs that are in the domain of outer function

anonymous · Answer

So if I wanted to the intersect of some sort, how would it look like?

anonymous · Answer

I find remembering an intersection better than the wording.

anonymous · Answer

ok but the intersection of domain of inner with what?  those whose outputs are in the domain of outer.  that will work

anonymous · Answer

?

anonymous · Answer

I do not understand. Would you mind writing it out with the symbols?