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Determine the vertical asymptotes and holes (removable points of discontinuity) of the rational function shown below. f(x) = x² - 12x + 20 x² 6x + 8

Determine the vertical asymptotes and holes (removable points of discontinuity) of the rational function shown below. f(x) = x² - 12x + 20 x² 6x + 8

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Determining the Vertical Asymptotes and Holes of a Rational Function To find the vertical asymptotes and holes (removable points of discontinuity) of the rational function: \[ f(x) = \frac{x^2 - 12x + 20}{x^2 - 6x + 8} \] Follow these steps: 1. **Factor both the numerator and denominator:** \[ f(x) = \frac{(x-2)(x-10)}{(x-2)(x-4)} \] 2. **Identify common factors in the numerator and denominator:** The factor \( (x-2) \) is common in both the numerator and denominator. 3. **Determine the holes and vertical asymptotes:** - **Holes:** These occur where there are common factors, meaning there is a "gap" in the graph. For this function, the hole is at \( x = 2 \). - **Vertical Asymptotes:** These occur where the denominator is zero (after accounting for holes), meaning the function approaches infinity. For this function, the vertical asymptote is at \( x = 4 \). ### Input Fields - **Holes:** Enter \( x = 2 \) if the function has a hole. - **Vertical Asymptotes:** Enter \( x = 4 \) if the function has a vertical asymptote. ### Example Input - **Holes:** [input box] - **Vertical Asymptotes:** x = 4 ✔ By understanding this process, students can effectively determine the points of discontinuity in rational functions.
Transcribed Image Text:### Determining the Vertical Asymptotes and Holes of a Rational Function To find the vertical asymptotes and holes (removable points of discontinuity) of the rational function: \[ f(x) = \frac{x^2 - 12x + 20}{x^2 - 6x + 8} \] Follow these steps: 1. **Factor both the numerator and denominator:** \[ f(x) = \frac{(x-2)(x-10)}{(x-2)(x-4)} \] 2. **Identify common factors in the numerator and denominator:** The factor \( (x-2) \) is common in both the numerator and denominator. 3. **Determine the holes and vertical asymptotes:** - **Holes:** These occur where there are common factors, meaning there is a "gap" in the graph. For this function, the hole is at \( x = 2 \). - **Vertical Asymptotes:** These occur where the denominator is zero (after accounting for holes), meaning the function approaches infinity. For this function, the vertical asymptote is at \( x = 4 \). ### Input Fields - **Holes:** Enter \( x = 2 \) if the function has a hole. - **Vertical Asymptotes:** Enter \( x = 4 \) if the function has a vertical asymptote. ### Example Input - **Holes:** [input box] - **Vertical Asymptotes:** x = 4 ✔ By understanding this process, students can effectively determine the points of discontinuity in rational functions.
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