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Math Help - range of rational quadratic function

  1. #1
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    range of rational quadratic function

    I want to show the range of y is R for all x. y=(2x-1)/(2x^2-4x+1)

    I made a quadratic equation and used the fact that the discriminant >=0 for real x. It was solved to y>=0.5(i-1).
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  2. #2
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    Re: range of rational quadratic function

    What is the meaning of y \ge .5(i-1)? How do you define inequalities in the complex plane?

    We want to show that for \frac{2x-1}{2x^2 - 4x + 1} = k, no matter what our choice of k is, there is always a real solution for x. Equation becomes

    2x - 1 = 2kx^2 - 4kx + k

    2kx^2 - (4k+2)x + (k+1) = 0

    The discriminant D is

    D = (4k+2)^2 - 4(2k)(k+1) = 8k^2 + 8k + 4, which is positive for all real k.
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  3. #3
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    Re: range of rational quadratic function

    That is the discriminant I got. I thought I had to factorise it. Is there nothing further to do?

    I am finding I can't make replies using IE8. Is this a common problem with this forum?
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