Posted by Sarah on .
Mathmate, I really appreciate all your help regarding my composite function question below but I'm still in need of your help. I don't understand how the range of x^2x+1 is R. Isn't this a parabola with a minimum for y? Also, I understand minimum and maximum values of sine but I don't know how to apply in my question. Again, many thanks for helping me out with this 'cause it's really bugging me.

Math 
MathMate,
You're right. The range of an odddegreed polynomial is ℝ but even degreed polynomials are limited on one side. It's a slip on my part.
In this case, you need to find the minimum (at the vertex) of f(x)=x²x+1 (1/2,3/4). So the range is [3/4,∞).
To find the minimum and maximum values (range) of g(x)=sin²(x)sin(x)+1,
you would use differentiation to find the derivative, g'(x)=2sin(x)cos(x)cos(x)
and equate g'(x) to zero to get
2sin(x)cos(x)cos(x)=0
cos(x)(2sin(x)1)=0
thus
cos(x)=0 or sin(x)=1/2
Solution set in [0,2π] is
{π/2, 3π/2, π/6, and 5π/6}
These are the possible locations for absolute minimum/maximum.
Evaluate g(x) at these points and determine the values of the absolute minimum and maximum. These are the limits of the range, since g(x) is a continuous function.
If you have not yet done differential calculus, you can draw the graph of g(x) and select points near which you can calculate a refined value of the absolute maximum and minimum.
Here's graph of the function g(x) between 0 and 2π.
http://img535.imageshack.us/img535/7599/1285459685.png 
Math 
Sarah,
Thanks very much for taking the time to help! Since I haven't done calculus, I'll try graphing.

Math 
MathMate,
You're welcome!
Post if you have other questions, or need more details.