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Math Help - Quotient of two functions

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    Quotient of two functions

    Determine the zeroes, and the domain. Write the equation for each asymptote. Then graph the function and estimate the range.

    g(x)= x/x^2-4 h(x)= x^2-4/x

    Thank you
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    Are the functions g(x) = \frac{x}{{x^2  - 4}}\,\& \,h(x) = \frac{{x^2  - 4}}{x}?

    Please learn some advanced LaTeX.
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    Forum Admin topsquark's Avatar
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    ... Or at least parenthesis.

    -Dan
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    yes Plato is right about the equations....sorry about that
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by johett View Post
    Determine the zeroes, and the domain. Write the equation for each asymptote. Then graph the function and estimate the range.

    g(x)= \frac{x}{x^2-4}
    First factor the numerator and denominator and see if anything cancels out.
    g(x) = \frac{x}{(x + 2)(x - 2)}

    No cancellations.

    So to find vertical asymptotes find out where the denominator is equal to 0. This gives x = 2 and x = -2 as vertical asymptotes.

    This also gives the domain as all real numbers except x = 2, and -2.

    As far as the zeros are concerned, solve
    g(x) = \frac{x}{(x + 2)(x - 2)} = 0

    This has a solution of x = 0, so there is your zero.

    Is there a horizontal asymptote? For that we need to see what the behavior of g(x) is for very large x. I think it is easy to see that as x goes to either plus or minus infinity that g(x) goes to 0. So there is a horizontal asymptote at y = 0.

    We do not have a slant asymptote because the degree of the numerator is not one more than the degree of the denominator.

    I think that about covers it. I'll leave you to graph it yourself.

    -Dan
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    Quote Originally Posted by johett View Post
    Determine the zeroes, and the domain. Write the equation for each asymptote. Then graph the function and estimate the range.

    g(x)= x/x^2-4 h(x)= x^2-4/x

    Thank you
    Hello,

    some remarks about the function h:

    1. h(x) = \frac{x^2-4}{x} = x-\frac4x = \frac1{g(x)}~,~x\ \in \ \mathbb{R} \setminus \{0\}

    Therefore: The zeros of g indicates the vertical asymptotes of h.
    The vertical asymptotes of g pass through the zeros of h.

    2. h has a slanted asymptote y = x and a vertical asymptote at x = 0

    3. h has 2 zeros: x = -2, x = 2

    4. The graph of h is drawn in red, the asymptotes in brown.
    The blue graph with it's green asymptotes is the graph of g.
    Attached Thumbnails Attached Thumbnails Quotient of two functions-gebr_ratfkt.gif  
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