Write a rational function that has at least two vertical asymptotes, one hole, and one horizontal asymptote.

Question

 Write a rational function that has at least two vertical asymptotes, one hole, and one horizontal asymptote.   Write the function in factored form and then write it in standard form. List the vertical asymptotes, the horizontal asymptote and the hole. Describe the end behavior of the function.

Answer 1

f(x) = 2(x - 1)(x² + 3)/[(x - 1)(x + 2)x].

This function has 2 vertical asymptotes x = 0 and x = - 2; 1 horizontal asymptote y = 2;

it has hole at x = 1, y = 2·4/3 = 8/3 or point (1, 8/3) is whole.

In standard form f(x) = 2(x³ - x² + 3x - 3)/(x³ + x² - 2x)