Removable vs. Vertical Discontinuities Explained

Full transcript

We're given the function f of x equals three x cubed plus two x squared, all divided by x squared minus x, and we need to figure out what kind of discontinuities it has. The first step is to factor both the numerator and the denominator completely so we can see what's really going on inside this function. The numerator factors into x squared times the quantity three x plus two, and the denominator factors into x times the quantity x minus one. Notice that one factor of x appears in both the top and the bottom, so those cancel out, leaving a hole in the graph at x equals zero — that's a removable discontinuity. After cancellation, the factor x minus one remains only in the denominator, which means the function shoots off to infinity at x equals one, giving us a true vertical asymptote. So the correct answer is Choice C: f has a removable discontinuity at x equals zero and a discontinuity due to a vertical asymptote at x equals one. Always factor first!