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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Question
Find all asymptotes that this function might have (vertical, and/or horizontal, and/or slanted).
If a horizontal asymptote exists, find whether the function intersects the horizontal asymptote. If it does, what is the point of intersection?
![The given mathematical expression is:
\[ f(x) = \frac{x^2 - 4}{x^2 - 4x} \]
This function is a rational expression where the numerator is \( x^2 - 4 \) and the denominator is \( x^2 - 4x \).
### Explanation:
- **Numerator:** \( x^2 - 4 \) can be factored as \( (x - 2)(x + 2) \).
- **Denominator:** \( x^2 - 4x \) can be factored as \( x(x - 4) \).
### Important Considerations:
- **Domain:** The function is undefined where the denominator is zero. Thus, solve \( x^2 - 4x = 0 \). Factoring gives \( x(x - 4) = 0 \), indicating the function is undefined at \( x = 0 \) and \( x = 4 \).
- **Holes and Asymptotes:**
- Check if any factors cancel out between the numerator and denominator to identify holes.
- Vertical asymptotes might occur at points where the denominator is zero, provided they do not cancel out with factors in the numerator.
Understanding this function involves analyzing its behavior at different values of \( x \) and identifying any discontinuities or asymptotic behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4931441c-10c9-4e0d-964b-11065e671b38%2Fdf47a732-0335-4bae-8cf4-33c9f8a53193%2F2e8shlb_processed.png&w=3840&q=75)
Transcribed Image Text:The given mathematical expression is:
\[ f(x) = \frac{x^2 - 4}{x^2 - 4x} \]
This function is a rational expression where the numerator is \( x^2 - 4 \) and the denominator is \( x^2 - 4x \).
### Explanation:
- **Numerator:** \( x^2 - 4 \) can be factored as \( (x - 2)(x + 2) \).
- **Denominator:** \( x^2 - 4x \) can be factored as \( x(x - 4) \).
### Important Considerations:
- **Domain:** The function is undefined where the denominator is zero. Thus, solve \( x^2 - 4x = 0 \). Factoring gives \( x(x - 4) = 0 \), indicating the function is undefined at \( x = 0 \) and \( x = 4 \).
- **Holes and Asymptotes:**
- Check if any factors cancel out between the numerator and denominator to identify holes.
- Vertical asymptotes might occur at points where the denominator is zero, provided they do not cancel out with factors in the numerator.
Understanding this function involves analyzing its behavior at different values of \( x \) and identifying any discontinuities or asymptotic behavior.
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