The functions f and g are defined as f(x) = 4x +3 and g(x) = 1-2x 91 and a) Find the domain of f, g, f+g, f-g, fg, ff, g (₁ b) Find (f+g)(x), (f− g)(x), (fg)(x), (ff)(x), (x), and (9)
The functions f and g are defined as f(x) = 4x +3 and g(x) = 1-2x 91 and a) Find the domain of f, g, f+g, f-g, fg, ff, g (₁ b) Find (f+g)(x), (f− g)(x), (fg)(x), (ff)(x), (x), and (9)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
ChapterP: Prerequisites
SectionP.7: A Library Of Parent Functions
Problem 47E
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![### Functions and Their Domains
The functions \( f \) and \( g \) are defined as:
\[ f(x) = 4x + 3 \]
\[ g(x) = 1 - 2x \]
#### Question a)
Find the domain of the following expressions: \( f \), \( g \), \( f + g \), \( f - g \), \( fg \), \( \frac{f}{g} \), and \( \frac{g}{f} \).
#### Question b)
Find the following compositions and products of functions:
\[ (f + g)(x) \]
\[ (f - g)(x) \]
\[ (fg)(x) \]
\[ (ff)(x) \]
\[ \left(\frac{f}{g}\right)(x) \]
\[ \left(\frac{g}{f}\right)(x) \]
#### Answer Box for Question a)
The domain of \( f \) is:
\[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \]
(Type your answer in interval notation.)
---
### Explanations
- **Function Definitions**:
- \( f(x) = 4x + 3 \) is a linear function.
- \( g(x) = 1 - 2x \) is also a linear function.
- **Domain of a Function**:
- The domain of a linear function like \( f(x) \) or \( g(x) \) is typically all real numbers, unless otherwise restricted.
To determine the domain of compositions and products of these functions, or when dealing with ratios \( \frac{f}{g} \) and \( \frac{g}{f} \), consider where the functions are undefined (such as divisions by zero).
For \( \frac{f}{g} \):
\[ \frac{4x + 3}{1 - 2x} \]
The function is undefined when \( g(x) = 0 \):
\[ 1 - 2x = 0 \Rightarrow x = \frac{1}{2} \]
Thus, the function \( \frac{f}{g} \) is undefined at \( x = \frac{1}{2} \).
For \( \frac{g}{f} \):
\[ \frac{1 - 2x}{4x + 3} \]
The function is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c22293a-3ae6-4ac2-9c37-c12acd82a0e8%2F13b2e53a-ade0-49c6-a5ec-28d0e1ef6796%2Ft2b7ym_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Functions and Their Domains
The functions \( f \) and \( g \) are defined as:
\[ f(x) = 4x + 3 \]
\[ g(x) = 1 - 2x \]
#### Question a)
Find the domain of the following expressions: \( f \), \( g \), \( f + g \), \( f - g \), \( fg \), \( \frac{f}{g} \), and \( \frac{g}{f} \).
#### Question b)
Find the following compositions and products of functions:
\[ (f + g)(x) \]
\[ (f - g)(x) \]
\[ (fg)(x) \]
\[ (ff)(x) \]
\[ \left(\frac{f}{g}\right)(x) \]
\[ \left(\frac{g}{f}\right)(x) \]
#### Answer Box for Question a)
The domain of \( f \) is:
\[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \]
(Type your answer in interval notation.)
---
### Explanations
- **Function Definitions**:
- \( f(x) = 4x + 3 \) is a linear function.
- \( g(x) = 1 - 2x \) is also a linear function.
- **Domain of a Function**:
- The domain of a linear function like \( f(x) \) or \( g(x) \) is typically all real numbers, unless otherwise restricted.
To determine the domain of compositions and products of these functions, or when dealing with ratios \( \frac{f}{g} \) and \( \frac{g}{f} \), consider where the functions are undefined (such as divisions by zero).
For \( \frac{f}{g} \):
\[ \frac{4x + 3}{1 - 2x} \]
The function is undefined when \( g(x) = 0 \):
\[ 1 - 2x = 0 \Rightarrow x = \frac{1}{2} \]
Thus, the function \( \frac{f}{g} \) is undefined at \( x = \frac{1}{2} \).
For \( \frac{g}{f} \):
\[ \frac{1 - 2x}{4x + 3} \]
The function is
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