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有理函数的关键特征
说一个有理函数在某点有一个「洞」，这是什么意思，又是什么样的代数结构导致了这种行为？
1. Introduction
Introduction
2. When a rational function has a
When a rational function has a
3. Sign charts and finding formulas for rational functions
Sign charts and finding formulas for rational functions
4. A zero that survives cancellation
For r(x) = (x^2 - 1)/(x^2 - 3x - 4), which numerator zero is a true zero of r?
5. The graphing device's missing hole
For f(x) = (x^3 - 6x^2 + 5x)/(x^2 - 1), what does cancelling (x - 1) reveal?
6. Full feature audit of a rational function
For h(x) = (x^2 - 8x + 12)/(x^2 - 3x - 18), what are its zero, vertical asymptote, hole and horizontal asymptote?
7. Constructing a sign chart
A rational function must have a vertical asymptote at x = -2, a zero at x = 1, a hole at x = 5, and a horizontal asymptote y = -3. Which construction works?
8. A cancelled factor leaves a hole
When a common factor cancels from a rational function, the cancelled input becomes a hole rather than a vertical asymptote.
9. Hole value after cancellation
For h(x) = (x^2 - 8x + 12)/(x^2 - 3x - 18), the factor (x - 6) cancels. What value does the reduced function approach at x = 6? Give the exact fraction.
10. Constructing a rational function from a graph
A rational function must have two holes, two vertical asymptotes, two zeros and a horizontal asymptote. Which construction achieves that?