Consider (the rule) f(x) = 2x+3 from real numbers to real numbers. Is this a function? Is this a bijection? Is f-¹ a fuction? Write f-¹ as a rule f-¹(x) = ....
Consider (the rule) f(x) = 2x+3 from real numbers to real numbers. Is this a function? Is this a bijection? Is f-¹ a fuction? Write f-¹ as a rule f-¹(x) = ....
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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can you please help with 4 and 5
![**Problem 4: Analysis of the Function f(x) = 2x + 3**
Consider the function \( f(x) = 2x + 3 \) mapping real numbers to real numbers.
- **Questions:**
- Is this a function?
- Is this a bijection?
- Is \( f^{-1} \) a function?
- Write \( f^{-1} \) as a rule: \( f^{-1}(x) = \ldots \)
This problem requires understanding properties of functions, specifically linear functions, and determining if an inverse function exists and its explicit form.
**Problem 5: Proving the Bijection of a Function**
Within the real numbers, define the sets:
- \( A = \{ x : 0 < x < 1 \} \)
- \( B = \{ x : 0 < x < \infty \} \)
Define the function \( f : A \to B \) by the rule:
\[ f(x) = \frac{x}{1 - x} \]
- **Show that \( f \) is a bijection.**
This involves demonstrating both injectivity (one-to-one) and surjectivity (onto). Understanding transformations and mapping between sets with given properties will be important in this proof.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4f94f1b-b461-4628-ade7-08bc846003b3%2F5cacf8cc-f35b-4570-ae35-715c68bb7a37%2F4tg3qur_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4: Analysis of the Function f(x) = 2x + 3**
Consider the function \( f(x) = 2x + 3 \) mapping real numbers to real numbers.
- **Questions:**
- Is this a function?
- Is this a bijection?
- Is \( f^{-1} \) a function?
- Write \( f^{-1} \) as a rule: \( f^{-1}(x) = \ldots \)
This problem requires understanding properties of functions, specifically linear functions, and determining if an inverse function exists and its explicit form.
**Problem 5: Proving the Bijection of a Function**
Within the real numbers, define the sets:
- \( A = \{ x : 0 < x < 1 \} \)
- \( B = \{ x : 0 < x < \infty \} \)
Define the function \( f : A \to B \) by the rule:
\[ f(x) = \frac{x}{1 - x} \]
- **Show that \( f \) is a bijection.**
This involves demonstrating both injectivity (one-to-one) and surjectivity (onto). Understanding transformations and mapping between sets with given properties will be important in this proof.
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