In rational functions, vertical asymptotes and holes represent values of $x$ where the function is undefined. Vertical asymptotes occur where the denominator of a simplified rational function equals zero. Holes, on the other hand, occur where a factor in both the numerator and denominator cancels out. Identifying these points helps understand the function's behavior and graph.
Match each term with its correct definition:
| Term | Definition |
|---|---|
| 1. Rational Function | A. A point where the function is undefined due to a common factor in the numerator and denominator |
| 2. Vertical Asymptote | B. A function that can be expressed as a fraction where both the numerator and denominator are polynomials |
| 3. Hole | C. A line $x = a$ where the function approaches infinity or negative infinity as $x$ approaches $a$ |
| 4. Simplified Rational Function | D. The function with common factors canceled out |
| 5. Undefined | E. A value that you cannot use for $x$, because it makes the function not exist |
A _________ function has vertical asymptotes where the _________ of the _________ rational function equals zero. A _________ occurs where a factor cancels out in both the _________ and _________.
Explain in your own words why it's important to simplify a rational function before identifying its vertical asymptotes and holes. Give an example of a rational function where not simplifying it first would lead to an incorrect identification of a vertical asymptote.
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