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,...
Related questions
Question
![Certainly! Here is the transcribed content for the educational website:
---
In this lesson, we will explore how to find the quotient of two functions and determine the domain of the resulting function.
Given the functions:
\[ f(x) = x^2 + x - 12 \]
\[ g(x) = x - 3 \]
**Task:**
1. **Find** \(\left(\frac{f}{g}\right)(x)\).
2. **Determine the domain of** \(\left(\frac{f}{g}\right)(x)\).
\[ \left(\frac{f}{g}\right)(x) = \boxed{\phantom{\rule{4cm}{0.4pt}}} \]
\[ \text{The domain of} \left(\frac{f}{g}\right)(x) \text{ is } x \ne \boxed{\phantom{\rule{2cm}{0.4pt}}} \]
---
**Explanation:**
To find \(\left(\frac{f}{g}\right)(x)\), we divide \( f(x) \) by \( g(x) \):
\[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 + x - 12}{x - 3} \]
Next, we factor the numerator \( x^2 + x - 12 \):
\[ x^2 + x - 12 = (x - 3)(x + 4) \]
So,
\[ \left(\frac{f}{g}\right)(x) = \frac{(x - 3)(x + 4)}{x - 3} \]
When simplifying, we must consider \( x \ne 3 \) (to avoid division by zero):
\[ \left(\frac{f}{g}\right)(x) = x + 4, \quad \text{for } x \ne 3 \]
Thus, the simplified function is:
\[ \left(\frac{f}{g}\right)(x) = x + 4 \]
**The domain of** \(\left(\frac{f}{g}\right)(x)\) **is** \( x \ne 3 \).
\[ \boxed{x + 4} \]
\[ \text{The domain of} \left(\frac{f}{g}\right)(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa78a88d7-60f0-4b9e-a272-6ff852667c68%2F9e8f1222-1baa-42a8-a97a-78e955c48f3a%2F2s7egke_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcribed content for the educational website:
---
In this lesson, we will explore how to find the quotient of two functions and determine the domain of the resulting function.
Given the functions:
\[ f(x) = x^2 + x - 12 \]
\[ g(x) = x - 3 \]
**Task:**
1. **Find** \(\left(\frac{f}{g}\right)(x)\).
2. **Determine the domain of** \(\left(\frac{f}{g}\right)(x)\).
\[ \left(\frac{f}{g}\right)(x) = \boxed{\phantom{\rule{4cm}{0.4pt}}} \]
\[ \text{The domain of} \left(\frac{f}{g}\right)(x) \text{ is } x \ne \boxed{\phantom{\rule{2cm}{0.4pt}}} \]
---
**Explanation:**
To find \(\left(\frac{f}{g}\right)(x)\), we divide \( f(x) \) by \( g(x) \):
\[ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 + x - 12}{x - 3} \]
Next, we factor the numerator \( x^2 + x - 12 \):
\[ x^2 + x - 12 = (x - 3)(x + 4) \]
So,
\[ \left(\frac{f}{g}\right)(x) = \frac{(x - 3)(x + 4)}{x - 3} \]
When simplifying, we must consider \( x \ne 3 \) (to avoid division by zero):
\[ \left(\frac{f}{g}\right)(x) = x + 4, \quad \text{for } x \ne 3 \]
Thus, the simplified function is:
\[ \left(\frac{f}{g}\right)(x) = x + 4 \]
**The domain of** \(\left(\frac{f}{g}\right)(x)\) **is** \( x \ne 3 \).
\[ \boxed{x + 4} \]
\[ \text{The domain of} \left(\frac{f}{g}\right)(
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 4 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education