Posted by Sarah on Saturday, September 25, 2010 at 10:30am.
If f(x)=sin(x) and g(x)=x^2x+1, find the domain of f(g(x)) and g(f(x)).
I was able to determine that:
f(g(x))= sin(x^2x+1) and
g(f(x))= sin(x)^2 sin(x) +1
Here is what I am thinking:
the range of f(x) is (x e R) and the range of g(x) is {x e R). Therefore the domain of f(g(x)) and g(f(x)) is {x e R)
the range of f(x) is {1=< y=< 1} and the range of g(x) is {y>= .75}. Therefore the range of g(x) and f(x) will be what satisfies both the functions, { .75=< y =< 1}
Is this correct? If not, how would I go about solving this problem, specifically the range? Thanks for your help.

Math  MathMate, Saturday, September 25, 2010 at 6:24pm
f(g(x))= sin(x^2x+1) and
g(f(x))= sin(x)^2 sin(x) +1
are correct.
The range of f(x)=sin(x) is [1,1].
The range of g(x)=x²x+1 is indeed ℝ.
However, the domain of both functions is ℝ.
So the domain of f(g(x)) ℝ and the range is still [1,1].
The domain of g(f(x)) is ℝ. However, since sin(x) is a periodic function, we just have to limit the search to [2π,2π].
Find the absolute minimum and absolute maximum on the interval [2π,2π] and that would be the range (all possible values of the function).
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