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.
Find any points of discontinuity for each rational function. Sketch the graph. Describe any vertical or horizontal asymptotes and any holes.
20. y = x^3 - 1/x^2 - 1
21. y = 2x^2 + 3/x^2 + 2
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