Find any points of discontinuity for each rational function. Sketch the graph. Describe any vertical or horizontal asymptotes and any holes.

Question

20. y = x^3 - 1/x^2 - 1
21. y = 2x^2 + 3/x^2 + 2

Answer 1

For the rational function y = x^3 - 1/x^2 - 1, the points of discontinuity occur when the denominator is equal to zero:

x^2 + 1 = 0
x^2 = -1
There are no real solutions for x in this case, so there are no points of discontinuity for this function.

To sketch the graph, note that the function has a vertical asymptote at x = 0 because the denominator approaches zero as x approaches zero. There are no horizontal or slant asymptotes. The function has no holes.

For the rational function y = 2x^2 + 3/x^2 + 2, the points of discontinuity occur when the denominator is equal to zero:

x^2 + 2 = 0
x^2 = -2
There are no real solutions for x in this case, so there are no points of discontinuity for this function.

To sketch the graph, note that the function has a vertical asymptote at x = 0 because the denominator approaches zero as x approaches zero. There are no horizontal or slant asymptotes. The function has no holes.