# Is it possible to find a rational function that has x-intercepts (-2,0) and (2,0), but has vertical asymptote x=1 and horizontal asymptote of y=0?

The horizontal asymptote and the x-intercepts parts stump me. If you can't reach y=0, then how can you get the x-intercepts?

And I don't know how to start finding a rational function that has asymptotes x=0 and y=x. It seems like I only have enough space to place a function in Q-2, Q-4.

Well, you can reach y = 0 at x = ±2, but you'll also reach it in the limit x to ± infinity.

You just write down the polynomila that has zeros ar ±2:

x^2 - 4

you divide this by a function whiich has a zero at x = 1: x - 1. You then get:

(x^2-4)/(x-1)

But this function does not go to zero if x goes to infinity. So, you divide this by a polynomial that has no zeroes to make the degree of the denominator larger than the degree of the numerator. You can, e.g. take that polynomial to be x^2 +1. Then you get:

(x^2-4)/[(x-1)(x^2+1)]

And I don't know how to start finding a rational function that has asymptotes x=0 and y=x. It seems like I only have enough space to place a function in Q-2, Q-4."

You can take:

1/x + x

## To find a rational function that has x-intercepts at (-2,0) and (2,0), but has a vertical asymptote at x=1 and a horizontal asymptote of y=0, follow these steps:

1. Start by considering the x-intercepts. Since the function crosses the x-axis at -2 and 2, we know that the numerator of the rational function should have factors of (x + 2) and (x - 2).

2. Next, let's focus on the vertical asymptote at x=1. To ensure that there is a vertical asymptote at x=1, include a factor of (x - 1) in the denominator of the rational function.

3. For the horizontal asymptote of y=0, we need to make the degree of the denominator at least one greater than the degree of the numerator. To do this, include an additional factor of (x - 1) in the denominator.

Putting it all together, the rational function that satisfies these conditions is:

f(x) = (x + 2)(x - 2) / [(x - 1)^2]

This function has x-intercepts at (-2,0) and (2,0), a vertical asymptote at x=1, and a horizontal asymptote of y=0.