6. For each of the following, is it a well-defined function? Explain. (a) (b) (d) a: R→Z a(x) = B(n) = B: Z→ {0, 1, -1} 1 -1 0 = x. if n is even, if n is positive, otherwise. (c) (Here Ro is the set of non-negative real numbers.) |-|: R → R≥o ||- {²₂ = -X 7: R R X |x| y(x) = = if x ≥ 0, if x ≤ 0.
6. For each of the following, is it a well-defined function? Explain. (a) (b) (d) a: R→Z a(x) = B(n) = B: Z→ {0, 1, -1} 1 -1 0 = x. if n is even, if n is positive, otherwise. (c) (Here Ro is the set of non-negative real numbers.) |-|: R → R≥o ||- {²₂ = -X 7: R R X |x| y(x) = = 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
Related questions
Question
![For each of the following, is it a well-defined function? Explain.
(a)
\[
\alpha : \mathbb{R} \to \mathbb{Z}
\]
\[
\alpha(x) = x.
\]
(b)
\[
\beta : \mathbb{Z} \to \{0, 1, -1\}
\]
\[
\beta(n) =
\begin{cases}
1 & \text{if } n \text{ is even,} \\
-1 & \text{if } n \text{ is positive,} \\
0 & \text{otherwise.}
\end{cases}
\]
(c)
(Here \(\mathbb{R}_{\geq 0}\) is the set of non-negative real numbers.)
\[
|-| : \mathbb{R} \to \mathbb{R}_{\geq 0}
\]
\[
|x| =
\begin{cases}
x & \text{if } x \geq 0, \\
-x & \text{if } x \leq 0.
\end{cases}
\]
(d)
\[
\gamma : \mathbb{R} \to \mathbb{R}
\]
\[
\gamma(x) = \frac{x}{|x|}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F382d69cf-ffe2-43c0-99c4-21bedf550518%2F740cdca2-77f7-416c-b972-0e23a3e3ca1c%2F2wl2bcd_processed.png&w=3840&q=75)
Transcribed Image Text:For each of the following, is it a well-defined function? Explain.
(a)
\[
\alpha : \mathbb{R} \to \mathbb{Z}
\]
\[
\alpha(x) = x.
\]
(b)
\[
\beta : \mathbb{Z} \to \{0, 1, -1\}
\]
\[
\beta(n) =
\begin{cases}
1 & \text{if } n \text{ is even,} \\
-1 & \text{if } n \text{ is positive,} \\
0 & \text{otherwise.}
\end{cases}
\]
(c)
(Here \(\mathbb{R}_{\geq 0}\) is the set of non-negative real numbers.)
\[
|-| : \mathbb{R} \to \mathbb{R}_{\geq 0}
\]
\[
|x| =
\begin{cases}
x & \text{if } x \geq 0, \\
-x & \text{if } x \leq 0.
\end{cases}
\]
(d)
\[
\gamma : \mathbb{R} \to \mathbb{R}
\]
\[
\gamma(x) = \frac{x}{|x|}.
\]
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