Given the function g(x)=2x^3-7x^2-8x/x^2-x-6 answer the following:?

a. What is the domain?

In set notation, the domain is { x ϵ ℝ | x ≠ -2 , x ≠ 3 }

That arises because the denominator can be factored as [(x + 2) * (x - 3)] , so the function is not defined at those values of x because they make the denominator equal to zero, and division by zero is undefined.

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b. Does this function have any vertical asymptotes? If yes, state there equation. If no, state why not.

Yes, the vertical asymptotes are where the function is undefined, and the equations are the lines x = -2 and x = 3 .

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c.Does this function have any horizontal asymptotes? If yes, state there equation. If no, state why not.

No, there are no horizontal asymptotes. Hmm ... when considering if there are horizontal asymptotes, you have to think about the behavior of the function as x gets really large positive and/or really large negative. You can do this precisely with limits and L'Hopital's rule. There's a thing in number theory that is helpful though, called 'big-O theory'. Basically it says that the end behavior of the function in the question will be the same as (x^3 / x^2) = x . [The lower degree terms don't matter when x gets really large positive or negative.] Since f(x) = x does not approach any specific number as x gets large positive or large negative, there are no horizontal asymptotes.

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d. Does this function have any oblique asymptotes? If yes, state there equation. If no, state why not.

Yes, it has 1 oblique asymptote, and the equation for that is y = 2x - 5

The following page describes how to find oblique asymptotes :

http://www.purplemath.com/modules/asymtote3.htm

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e. Find the zeros and y intercept of the graph of this function.

y-intercept is when x = 0 , so put that in to get y = 0 .

x-intercepts happen when the numerator equals zero.

2x^3 - 7x^2 - 8x = 0

x * (2x^2 - 7x - 8) = 0

So one x-intercept is at x = 0 (by the principal of zero products)

Need to find the zeros of (2x^2 - 7x - 8) , and I don't see an easy factorization of that, so I'll use the quadratic formula with a = 2 , b = - 7 , c = - 8 to get the roots

x = [(7 - √113) / 4] ≃ - 0.91

and

x = [(7 + √113) / 4] ≃ 4.41

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Here's the graph of the situation :

(coordinate axes are on different scales)

http://s1164.photobucket.com/user/iago9/media/math...