Suppose that the functions and s are defined for all real numbers x as follows. r r(x)=x-6 s(x) = 3x+2 Write the expressions for (r+s) (x) and (-s) (x) and evaluate (r.s)(1).
Suppose that the functions and s are defined for all real numbers x as follows. r r(x)=x-6 s(x) = 3x+2 Write the expressions for (r+s) (x) and (-s) (x) and evaluate (r.s)(1).
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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Question
![### Problem Statement
Suppose that the functions \( r \) and \( s \) are defined for all real numbers \( x \) as follows.
\[
r(x) = x - 6
\]
\[
s(x) = 3x + 2
\]
### Tasks
1. Write the expressions for \( (r + s)(x) \) and \( (r - s)(x) \).
2. Evaluate \( (r \cdot s)(1) \).
#### Step-by-Step Solution
##### 1. Finding \( (r + s)(x) \)
To find \( (r + s)(x) \), simply add the two functions together:
\[
(r + s)(x) = r(x) + s(x)
\]
Substituting the given functions:
\[
(r + s)(x) = (x - 6) + (3x + 2)
\]
Combine like terms:
\[
(r + s)(x) = x + 3x - 6 + 2
\]
\[
(r + s)(x) = 4x - 4
\]
##### 2. Finding \( (r - s)(x) \)
To find \( (r - s)(x) \), subtract the function \( s(x) \) from \( r(x) \):
\[
(r - s)(x) = r(x) - s(x)
\]
Substituting the given functions:
\[
(r - s)(x) = (x - 6) - (3x + 2)
\]
Distribute the negative sign:
\[
(r - s)(x) = x - 6 - 3x - 2
\]
Combine like terms:
\[
(r - s)(x) = x - 3x - 6 - 2
\]
\[
(r - s)(x) = -2x - 8
\]
##### 3. Evaluating \( (r \cdot s)(1) \)
To find \( (r \cdot s)(1) \), multiply the values of \( r(x) \) and \( s(x) \) at \( x = 1 \):
\[
(r \cdot s)(1) = r(1) \cdot s(1)
\]
Calculate \( r(1) \):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a47fda9-061f-451e-94e6-6988f38e6732%2F1f8abc86-f9a3-4a6d-b9e1-e9e94780e5cf%2Fp2f0dq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Suppose that the functions \( r \) and \( s \) are defined for all real numbers \( x \) as follows.
\[
r(x) = x - 6
\]
\[
s(x) = 3x + 2
\]
### Tasks
1. Write the expressions for \( (r + s)(x) \) and \( (r - s)(x) \).
2. Evaluate \( (r \cdot s)(1) \).
#### Step-by-Step Solution
##### 1. Finding \( (r + s)(x) \)
To find \( (r + s)(x) \), simply add the two functions together:
\[
(r + s)(x) = r(x) + s(x)
\]
Substituting the given functions:
\[
(r + s)(x) = (x - 6) + (3x + 2)
\]
Combine like terms:
\[
(r + s)(x) = x + 3x - 6 + 2
\]
\[
(r + s)(x) = 4x - 4
\]
##### 2. Finding \( (r - s)(x) \)
To find \( (r - s)(x) \), subtract the function \( s(x) \) from \( r(x) \):
\[
(r - s)(x) = r(x) - s(x)
\]
Substituting the given functions:
\[
(r - s)(x) = (x - 6) - (3x + 2)
\]
Distribute the negative sign:
\[
(r - s)(x) = x - 6 - 3x - 2
\]
Combine like terms:
\[
(r - s)(x) = x - 3x - 6 - 2
\]
\[
(r - s)(x) = -2x - 8
\]
##### 3. Evaluating \( (r \cdot s)(1) \)
To find \( (r \cdot s)(1) \), multiply the values of \( r(x) \) and \( s(x) \) at \( x = 1 \):
\[
(r \cdot s)(1) = r(1) \cdot s(1)
\]
Calculate \( r(1) \):
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