Determine the zeroes, and the domain. Write the equation for each asymptote. Then graph the function and estimate the range.
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
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.
some remarks about the function h:
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.