March 29, 2017

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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^2-x+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 - ,

    You're right. The range of an odd-degreed 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)=x²-x+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
    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.

  • Math - ,

    Thanks very much for taking the time to help! Since I haven't done calculus, I'll try graphing.

  • Math - ,

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

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