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Determine the domain of the following function: f(x)=sqrt x-7/x^2-5x-6

I think the domain is
[7,all real numbers)

Can you please tell me if this is correct?

  • PRE-CALC -

    You have to be more generous with your parentheses. As is the function does not seem to be properly defined.
    It could be
    f(x) = sqrt( (x-7)/(x²-5x-6) )
    g(x) = sqrt(x-7) / (x²-5x-6) )

    I assume it is f(x). If it is g(x), you can proceed along the same lines, and post your results for confirmation.

    f(x) can be rewritten as:
    f(x) = sqrt( (x-7)/((x-2)(x-3)) )
    which tells us that
    1. at x<7, the numerator becomes negative.
    2. The denominator is a parabola which has zeroes at x=2 and x=3.
    3. The denominator is negative between x=2 and x=3. It is positive elsewhere.

    From 1 and 3, we conclude that the fraction inside the square radical is
    1. negative when x<2,
    2. positive when 2<x<3
    3. negative when 3<x<7
    4. positive when x>7

    Also, there are two vertical asymptotes at x=2 and x=3 which should be removed from the domain of f(x).

    Therefore the domain of f(x) is:

  • PRE-CALC -

    Actually, there are no parenthesis. The problem is how I wrote it but
    sqrt x-7 is over x^2 -5x - 6

    I factored it to (x-6) (x+1)

  • PRE-CALC -

    " Actually, there are no parenthesis. The problem is how I wrote it but
    sqrt x-7 is over x^2 -5x - 6 "
    You have to be more generous with parentheses. Whenever a fraction is transcribed to a single line, you will need to insert parentheses around the numerator AND the denominator to avoid ambiguity.
    From what you described, you would write the expression as
    sqrt(x-7)/(x^2 -5x - 6)
    otherwise it can be (unlikely) interpreted as
    sqrt(x) -7/x² -5x -6
    which is not a normal function.

    Back to the function:
    f(x) = sqrt(x-7)/(x^2 -5x - 6)
    you have already correctly identified the donmain [7,∞).
    The only other considerations required would be to exclude the two singular points (x=2 and x=3) from the domain.
    Post your results for checking if you wish.

  • PRE-CALC -

    Thanks I didn't realize how to type it when one is over the other. I will keep it in mind.

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