For each of the following, is it a well-defined function? Explain. (a) (b) (d) a: R→ Z a(x) = x. 3: Z {0, 1,-1} { B(n) = -1 (c) (Here Ro is the set of non-negative real numbers. ) | − | : R → R≥o - >0 |x| if n is even, if n is positive, otherwise. X -X 7: R R y(x) = 8 if x ≥ 0, if x ≤ 0.
For each of the following, is it a well-defined function? Explain. (a) (b) (d) a: R→ Z a(x) = x. 3: Z {0, 1,-1} { B(n) = -1 (c) (Here Ro is the set of non-negative real numbers. ) | − | : R → R≥o - >0 |x| if n is even, if n is positive, otherwise. X -X 7: R R y(x) = 8 if x ≥ 0, if x ≤ 0.
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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![### Evaluating the Well-Definedness of Functions
This section explores whether the following mappings can be considered well-defined functions. A function is well-defined if each element from the domain is mapped to exactly one element in the codomain.
#### (a) Function \( \alpha \)
- **Mapping:** \( \alpha : \mathbb{R} \to \mathbb{Z} \)
- **Definition:** \( \alpha(x) = x \)
**Explanation:** This mapping is not well-defined because real numbers (\(\mathbb{R}\)) generally have fractional parts, which cannot belong to the set of integers (\(\mathbb{Z}\)). Only real numbers that are integers can be mapped directly.
#### (b) Function \( \beta \)
- **Mapping:** \( \beta : \mathbb{Z} \to \{0, 1, -1\} \)
- **Definition:**
\[
\beta(n) =
\begin{cases}
1 & \text{if } n \text{ is even,} \\
-1 & \text{if } n \text{ is positive,} \\
0 & \text{otherwise.}
\end{cases}
\]
**Explanation:** This mapping is not well-defined. Consider \( n = 2 \); it satisfies both \( n \) being even and positive, which would map it to both 1 and -1, contradicting the well-defined property of a function.
#### (c) Absolute Value Function
- **Mapping:** \( | \cdot | : \mathbb{R} \to \mathbb{R}_{\geq 0} \)
- **Definition:**
\[
|x| =
\begin{cases}
x & \text{if } x \geq 0, \\
-x & \text{if } x \leq 0.
\end{cases}
\]
**Explanation:** This mapping is well-defined. Each real number \( x \) has one unique non-negative \( y \) such that \( y = |x| \).
#### (d) Function \( \gamma \)
- **Mapping:** \( \gamma : \mathbb{R} \to \mathbb{R} \)
- **Definition:** \( \gamma(x) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed3eef2a-70e8-485e-b91d-563de65921b3%2Ffecbce0a-9ac6-498a-a01d-860e968fa2ed%2Fxf37ya_processed.png&w=3840&q=75)
Transcribed Image Text:### Evaluating the Well-Definedness of Functions
This section explores whether the following mappings can be considered well-defined functions. A function is well-defined if each element from the domain is mapped to exactly one element in the codomain.
#### (a) Function \( \alpha \)
- **Mapping:** \( \alpha : \mathbb{R} \to \mathbb{Z} \)
- **Definition:** \( \alpha(x) = x \)
**Explanation:** This mapping is not well-defined because real numbers (\(\mathbb{R}\)) generally have fractional parts, which cannot belong to the set of integers (\(\mathbb{Z}\)). Only real numbers that are integers can be mapped directly.
#### (b) Function \( \beta \)
- **Mapping:** \( \beta : \mathbb{Z} \to \{0, 1, -1\} \)
- **Definition:**
\[
\beta(n) =
\begin{cases}
1 & \text{if } n \text{ is even,} \\
-1 & \text{if } n \text{ is positive,} \\
0 & \text{otherwise.}
\end{cases}
\]
**Explanation:** This mapping is not well-defined. Consider \( n = 2 \); it satisfies both \( n \) being even and positive, which would map it to both 1 and -1, contradicting the well-defined property of a function.
#### (c) Absolute Value Function
- **Mapping:** \( | \cdot | : \mathbb{R} \to \mathbb{R}_{\geq 0} \)
- **Definition:**
\[
|x| =
\begin{cases}
x & \text{if } x \geq 0, \\
-x & \text{if } x \leq 0.
\end{cases}
\]
**Explanation:** This mapping is well-defined. Each real number \( x \) has one unique non-negative \( y \) such that \( y = |x| \).
#### (d) Function \( \gamma \)
- **Mapping:** \( \gamma : \mathbb{R} \to \mathbb{R} \)
- **Definition:** \( \gamma(x) =
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