Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Use direct proofs
![**Mathematical Statement:**
If \( x \) is even, then \( 7x + 9 \) is odd.
**Explanation:**
This statement expresses a mathematical condition and conclusion:
- **Condition:** \( x \) is an even number.
- **Conclusion:** When \( 7 \times x \) (where \( x \) is even) is computed and 9 is added to the result, the final value is an odd number.
**Reasoning:**
To understand why this is true, consider the properties of even and odd numbers:
1. **Even Numbers:** Any even number can be expressed as \( 2k \), where \( k \) is an integer.
2. **Odd Numbers:** An odd number can be expressed as \( 2m + 1 \), where \( m \) is an integer.
When \( x \) is even, it can be written as \( x = 2k \). Substituting this into the equation \( 7x + 9 \), we get:
\[
7(2k) + 9 = 14k + 9
\]
Breaking this down:
- \( 14k \) is an even number because it is 2 times an integer.
- Adding 9 (an odd number) to an even number results in an odd number.
Thus, \( 7x + 9 \) is confirmed to be odd when \( x \) is even.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F421c078f-df9f-435c-ad8c-92936aa1531e%2Feba0fc9c-6d66-42a9-81b3-b906369239a0%2Fuxuwlk_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematical Statement:**
If \( x \) is even, then \( 7x + 9 \) is odd.
**Explanation:**
This statement expresses a mathematical condition and conclusion:
- **Condition:** \( x \) is an even number.
- **Conclusion:** When \( 7 \times x \) (where \( x \) is even) is computed and 9 is added to the result, the final value is an odd number.
**Reasoning:**
To understand why this is true, consider the properties of even and odd numbers:
1. **Even Numbers:** Any even number can be expressed as \( 2k \), where \( k \) is an integer.
2. **Odd Numbers:** An odd number can be expressed as \( 2m + 1 \), where \( m \) is an integer.
When \( x \) is even, it can be written as \( x = 2k \). Substituting this into the equation \( 7x + 9 \), we get:
\[
7(2k) + 9 = 14k + 9
\]
Breaking this down:
- \( 14k \) is an even number because it is 2 times an integer.
- Adding 9 (an odd number) to an even number results in an odd number.
Thus, \( 7x + 9 \) is confirmed to be odd when \( x \) is even.
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