Holes in rational functions occur when

Holes in rational functions occur when

• A. A common factor cancels between numerator and denominator
• B. The function has no domain
• C. The graph has intercepts only
• D. The function is continuous everywhere

Answer: (A)

Holes occur at x-values where the numerator and denominator share a common factor. That common factor cancels, so the simplified form would be defined at that x, but the original expression is not because the denominator is zero there. Graphically, you see a point missing on the curve—a hole—while the rest of the graph follows the shortened, simplified expression. This is a removable discontinuity. If the denominator vanishes without a matching factor in the numerator, you’d get a vertical asymptote instead, not a hole. So the situation described is exactly when a common factor cancels between numerator and denominator.

Holes occur at x-values where the numerator and denominator share a common factor. That common factor cancels, so the simplified form would be defined at that x, but the original expression is not because the denominator is zero there. Graphically, you see a point missing on the curve—a hole—while the rest of the graph follows the shortened, simplified expression. This is a removable discontinuity. If the denominator vanishes without a matching factor in the numerator, you’d get a vertical asymptote instead, not a hole. So the situation described is exactly when a common factor cancels between numerator and denominator.