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

g(x)=$\displaystyle x/x^2-4$ h(x)=$\displaystyle x^2-4/x$

Thank you

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- Jan 8th 2008, 03:22 PMjohettQuotient 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)=$\displaystyle x/x^2-4$ h(x)=$\displaystyle x^2-4/x$

Thank you - Jan 8th 2008, 03:28 PMPlato
Are the functions $\displaystyle g(x) = \frac{x}{{x^2 - 4}}\,\& \,h(x) = \frac{{x^2 - 4}}{x}$?

Please learn some advanced LaTeX. - Jan 8th 2008, 04:12 PMtopsquark
... Or at least parenthesis.

-Dan - Jan 8th 2008, 04:52 PMjohett
yes Plato is right about the equations....sorry about that

- Jan 8th 2008, 05:59 PMtopsquark
First factor the numerator and denominator and see if anything cancels out.

$\displaystyle g(x) = \frac{x}{(x + 2)(x - 2)}$

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

$\displaystyle g(x) = \frac{x}{(x + 2)(x - 2)} = 0$

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 - Jan 8th 2008, 09:09 PMearboth
Hello,

some remarks about the function h:

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

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.