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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![### Problem 2
**Task:**
Find the domain of the rational function and graph the function (show all work when graphing).
**Function:**
\[ f(x) = \frac{x}{x^2 - 16} \]
---
**Solution:**
1. **Find the Domain:**
- The domain of a rational function excludes values that make the denominator equal to zero.
- Set the denominator equal to zero and solve for \( x \):
\[
x^2 - 16 = 0
\]
\[
x^2 = 16
\]
\[
x = \pm 4
\]
- Thus, the values \( x = 4 \) and \( x = -4 \) make the denominator zero.
- Therefore, the domain of \( f(x) \) is:
\[
\text{Domain: } \{ x \in \mathbb{R} \mid x \neq -4, x \neq 4 \}
\]
2. **Graph the Function:**
- To graph the function, plot several points for different values of \( x \).
- Identify vertical asymptotes at \( x = 4 \) and \( x = -4 \), where the function is undefined.
- Determine the behavior of the function as \( x \) approaches these asymptotes from both the left and right.
- Determine any horizontal asymptotes by analyzing the degrees of the polynomial in the numerator and the denominator.
- As \( x \) approaches large positive or negative values, because the degree of the numerator is 1 (which is less than the degree of the denominator, which is 2), the horizontal asymptote is \( y = 0 \).
- Plot additional points to create the curve of the function. Ensure to show all steps in the graphing process, including calculating values at various points and illustrating the asymptotic behavior.
**Note:** When graphing, start by plotting key points (such as intercepts), then move on to identify behavior near asymptotes, followed by plotting several additional points to accurately depict the curve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd59633dd-b674-4b8b-8a6e-1c7bb3ffab4f%2F1ba5bd69-0000-40f1-b0bb-a2bf29caada8%2F2px32c2i.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 2
**Task:**
Find the domain of the rational function and graph the function (show all work when graphing).
**Function:**
\[ f(x) = \frac{x}{x^2 - 16} \]
---
**Solution:**
1. **Find the Domain:**
- The domain of a rational function excludes values that make the denominator equal to zero.
- Set the denominator equal to zero and solve for \( x \):
\[
x^2 - 16 = 0
\]
\[
x^2 = 16
\]
\[
x = \pm 4
\]
- Thus, the values \( x = 4 \) and \( x = -4 \) make the denominator zero.
- Therefore, the domain of \( f(x) \) is:
\[
\text{Domain: } \{ x \in \mathbb{R} \mid x \neq -4, x \neq 4 \}
\]
2. **Graph the Function:**
- To graph the function, plot several points for different values of \( x \).
- Identify vertical asymptotes at \( x = 4 \) and \( x = -4 \), where the function is undefined.
- Determine the behavior of the function as \( x \) approaches these asymptotes from both the left and right.
- Determine any horizontal asymptotes by analyzing the degrees of the polynomial in the numerator and the denominator.
- As \( x \) approaches large positive or negative values, because the degree of the numerator is 1 (which is less than the degree of the denominator, which is 2), the horizontal asymptote is \( y = 0 \).
- Plot additional points to create the curve of the function. Ensure to show all steps in the graphing process, including calculating values at various points and illustrating the asymptotic behavior.
**Note:** When graphing, start by plotting key points (such as intercepts), then move on to identify behavior near asymptotes, followed by plotting several additional points to accurately depict the curve.
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