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Math Help - Find the range of a function

  1. #1
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    Find the range of a function

    How do you algebraicly find the range of a function. I can tell what the range is supposed to be by looking at the graph, but I cannot show it using algebra.
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  2. #2
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    Quote Originally Posted by redier
    How do you algebraicly find the range of a function. I can tell what the range is supposed to be by looking at the graph, but I cannot show it using algebra.
    Excellent question. It used to bother me too, until I developed a trick.
    Consider a function,
    y=f(x)
    The range are all the values of y such that there exists an x in the domain such as,
    y=f(x).
    ---
    Example,
    Find the range of,
    y=\frac{1}{1+x^2}
    The domain is all real numbers.
    Thus, for what values of y does the equation have a solution.
    Say,
    y=\frac{1}{1+x^2}
    Then,
    \frac{1}{y}=1+x^2
    Thus,
    \frac{1}{y}-1=x^2
    Note this equation can only have a solution when the Left Hand side is non-negative. That is,
    \frac{1}{y}-1\geq 0.
    Thus,
    \frac{1}{y}\geq 1
    If you solve this inequality (details omitted) you have that,
    0<y\leq 1
    ---
    As an excersce,
    Find the range of,
    y=\sin x+\cos x
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