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Thread: undefined function.....

  1. #1
    Aug 2005

    undefined function.....

    Hi All,

    If I have a function: f(x) = X^2 + 10/(X-2)

    and X is 2, the second part of the equation is undefined, correct?
    If I want to find the maximum domain and range of the function, how do I
    handle this?
    Do I just ignore the undefined section because of the defined component?

    Thanks for your help.

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  2. #2
    Senior Member
    Jun 2005
    If one part of the formula is undefined at X=2 then it is likely that the function is undefined there. You should however be careful to check whether the formula can be simplified. For example, (X^2-4)/(X-2) is undefined at 2, but you can extract a factor of X-2 from numerator and denominator to get X+2, which is defined at X=2. This is a very simple example of a removable singularity.
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  3. #3
    MHF Contributor
    Apr 2005
    No, you cannot do that. The function f(x) is the whole [x^2 +10/(x-2)].
    You cannot separate the two terms of the [x^2 +10/(x-2)] and still call the separated terms as f(x).

    When x=2, the 10/(x-2) is undefined while the x^2 is defined. That means the whole [x^2 +10/(x-2)], which is f(x), is not defined.

    You want to find the maximum domain and range of f(x).
    Well, the domain of f(x) is all the values of x where f(x) is defined. In the whole set of real numbers, f(x) is defined, except when x=2, so
    Domain is the whole set of real numbers of x, except x=2.

    The range of f(x) is the set of real numbers that what the domain can make of f(x).
    We know that x=2 is not part of the domain, so f(2) is not part of the range.
    What is f(2)?
    f(2) = 2^2 +10/(2-2) = 4 + (undefined, or, say, infinity) = say, infinity.

    Well, in any case, infinity is not part of any range of any function because infinity is not a real number.

    For all the other real numbers value of x, all the real numbers value of y, including y=0, are covered.
    Therefore, the range of f(x) is all real numbers.
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