Mathematics Algebra

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Question:
If the expression (x^2 + 2x +4)/(x^2 - 2x - 4) has a value between (1/3) and 3 for all real values of x, determine the range of the expression [9.(3)^x + 6.(3)^2x + 4]/[ 9.(3)^2x - 6.(3)^x +4


My thoughts on the question:
I can see that if we substitute x=[3.(3)^x] we can turn the first expression into the one ,which we are told to find the range.But I have no idea on going further..

  • Mathematics Algebra -

    But (x^2 + 2x +4)/(x^2 - 2x - 4) does not always fit into the interval [1/3,3]. If x is very close to 1±√5 then the value is arbitrarily large

    f(3.2) = -129

    plus, you have a typo. I think you mean
    [9.(3)^2x + 6.(3)^x + 4]/[ 9.(3)^2x - 6.(3)^x +4]

    using u = 3.3^x
    That becomes (u^2+2u+4)/(u^2-2u-4)

    so, I guess its range is [1/3,3]

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