Find the domain, vertical asymptotes, and horizontal asymptotes of the function. X f(x) = x2-49 Enter the domain in interval notation. To enter ∞, type infinity. To enter U, type U. Domain:
Find the domain, vertical asymptotes, and horizontal asymptotes of the function. X f(x) = x2-49 Enter the domain in interval notation. To enter ∞, type infinity. To enter U, type U. Domain:
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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![**Finding the Domain and Asymptotes of Rational Functions**
---
**Objective:**
Determine the domain, vertical asymptotes, and horizontal asymptotes of the given function.
---
**Given Function:**
\[ f(x) = \frac{x}{x^2 - 49} \]
---
**Instructions:**
1. **Domain:** The domain of a function includes all the possible values of \( x \) for which the function is defined. In this case, identify the values of \( x \) that make the denominator zero since these will be excluded from the domain.
2. **Vertical Asymptotes:** Vertical asymptotes occur where the function goes to infinity, which usually happens at the values of \( x \) where the denominator is zero (excluding any holes created by canceled factors).
3. **Horizontal Asymptotes:** Horizontal asymptotes describe the behavior of the function as \( x \) approaches positive or negative infinity. These can be determined by comparing the degrees of the polynomial in the numerator and the denominator.
---
**Steps:**
1. **Find the Domain in Interval Notation:**
- Set the denominator equal to zero and solve for \( x \).
- Consider the values of \( x \) where the denominator is non-zero to determine the domain.
2. **Enter the domain in interval notation:**
- To enter \(\infty\), type "infinity".
- To enter \( \cup \) (union), type "U".
3. **Identify Vertical Asymptotes:**
- Analyze the factored form of the denominator.
4. **Identify Horizontal Asymptotes:**
- Compare the highest powers of \( x \) in the numerator and the denominator.
---
**Result:**
- **Domain:** Enter the domain using the following input field.
\`Domain: [ \_\_\_\_ ]\`
---
**Notes:**
- Exclude specific \( x \) values from the domain where the denominator equals zero to avoid undefined expression.
- Simplify the function as necessary to identify any canceled factors indicating holes rather than vertical asymptotes.
- For horizontal asymptotes, use the degrees comparison rule:
- If degrees are equal, \( y = \frac{leading\ coefficient\ of\ numerator}{leading\ coefficient\ of\ denominator} \)
- If the degree of the numerator is less than the denominator, \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22396abd-1dcc-4b0d-9124-fd12e74a7cb7%2F3e7c18fc-daf0-4fb6-8eb0-765d806a2804%2F0jd1mdi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Domain and Asymptotes of Rational Functions**
---
**Objective:**
Determine the domain, vertical asymptotes, and horizontal asymptotes of the given function.
---
**Given Function:**
\[ f(x) = \frac{x}{x^2 - 49} \]
---
**Instructions:**
1. **Domain:** The domain of a function includes all the possible values of \( x \) for which the function is defined. In this case, identify the values of \( x \) that make the denominator zero since these will be excluded from the domain.
2. **Vertical Asymptotes:** Vertical asymptotes occur where the function goes to infinity, which usually happens at the values of \( x \) where the denominator is zero (excluding any holes created by canceled factors).
3. **Horizontal Asymptotes:** Horizontal asymptotes describe the behavior of the function as \( x \) approaches positive or negative infinity. These can be determined by comparing the degrees of the polynomial in the numerator and the denominator.
---
**Steps:**
1. **Find the Domain in Interval Notation:**
- Set the denominator equal to zero and solve for \( x \).
- Consider the values of \( x \) where the denominator is non-zero to determine the domain.
2. **Enter the domain in interval notation:**
- To enter \(\infty\), type "infinity".
- To enter \( \cup \) (union), type "U".
3. **Identify Vertical Asymptotes:**
- Analyze the factored form of the denominator.
4. **Identify Horizontal Asymptotes:**
- Compare the highest powers of \( x \) in the numerator and the denominator.
---
**Result:**
- **Domain:** Enter the domain using the following input field.
\`Domain: [ \_\_\_\_ ]\`
---
**Notes:**
- Exclude specific \( x \) values from the domain where the denominator equals zero to avoid undefined expression.
- Simplify the function as necessary to identify any canceled factors indicating holes rather than vertical asymptotes.
- For horizontal asymptotes, use the degrees comparison rule:
- If degrees are equal, \( y = \frac{leading\ coefficient\ of\ numerator}{leading\ coefficient\ of\ denominator} \)
- If the degree of the numerator is less than the denominator, \(
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