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 8: Analysis of a Rational Function
Consider the rational function \( f(x) = \frac{2x^2 + 5x - 3}{x^2 - 9} \).
#### (a) Finding the Domain of \( f(x) \)
Find the domain of \( f(x) \). Show your work!
#### (b) Determining the Zeros of \( f(x) \)
Find all zeros of \( f(x) \). Show your work!
#### (c) Asymptotes of \( f(x) \)
Find all horizontal and vertical asymptotes of \( f(x) \). Show your work!
#### (d) Graphing \( f(x) \)
Sketch the graph of \( f(x) \).
---
### Solutions
#### (a) Finding the Domain of \( f(x) \)
To find the domain of \( f(x) \), we need to determine the values of \( x \) for which the function is defined. Since \( f(x) \) is a rational function, it is defined for all real numbers except where the denominator is zero.
The denominator of \( f(x) \) is:
\[ x^2 - 9 \]
Set the denominator equal to zero and solve for \( x \):
\[ x^2 - 9 = 0 \]
\[ (x - 3)(x + 3) = 0 \]
Thus, \( x = 3 \) and \( x = -3 \) are the points where the denominator is zero. Therefore, the domain of \( f(x) \) is:
\[ \text{Domain: } x \in \mathbb{R} \setminus \{-3, 3\} \]
#### (b) Determining the Zeros of \( f(x) \)
To find the zeros of \( f(x) \), set the numerator equal to zero and solve for \( x \):
\[ 2x^2 + 5x - 3 = 0 \]
Solve the quadratic equation:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = 5 \), and \( c = -3 \):
\[ x = \frac{-5 \pm \sqrt{25 + 24}}{4} \]
\[ x = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3ea6570-68e1-449f-84f9-2e2550697ea4%2F5f8e4543-b1b8-4e5e-88be-a6e20bb2ed82%2Ftcp7uuq.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 8: Analysis of a Rational Function
Consider the rational function \( f(x) = \frac{2x^2 + 5x - 3}{x^2 - 9} \).
#### (a) Finding the Domain of \( f(x) \)
Find the domain of \( f(x) \). Show your work!
#### (b) Determining the Zeros of \( f(x) \)
Find all zeros of \( f(x) \). Show your work!
#### (c) Asymptotes of \( f(x) \)
Find all horizontal and vertical asymptotes of \( f(x) \). Show your work!
#### (d) Graphing \( f(x) \)
Sketch the graph of \( f(x) \).
---
### Solutions
#### (a) Finding the Domain of \( f(x) \)
To find the domain of \( f(x) \), we need to determine the values of \( x \) for which the function is defined. Since \( f(x) \) is a rational function, it is defined for all real numbers except where the denominator is zero.
The denominator of \( f(x) \) is:
\[ x^2 - 9 \]
Set the denominator equal to zero and solve for \( x \):
\[ x^2 - 9 = 0 \]
\[ (x - 3)(x + 3) = 0 \]
Thus, \( x = 3 \) and \( x = -3 \) are the points where the denominator is zero. Therefore, the domain of \( f(x) \) is:
\[ \text{Domain: } x \in \mathbb{R} \setminus \{-3, 3\} \]
#### (b) Determining the Zeros of \( f(x) \)
To find the zeros of \( f(x) \), set the numerator equal to zero and solve for \( x \):
\[ 2x^2 + 5x - 3 = 0 \]
Solve the quadratic equation:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = 5 \), and \( c = -3 \):
\[ x = \frac{-5 \pm \sqrt{25 + 24}}{4} \]
\[ x = \
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