Let f: R → R be a function, let function g: (-∞, 0) U (0, ∞) → R be defined by g(x) = 1, and let function h: [0, ∞) → R be defined by h(x)=√x. A. In terms of f(x), find an expression for the functions (gof)(x), (hof)(x), (gohof)(x), (hogof)(x). B. Henceforth, let P be the collection of all x for which f(x) is positive, let N be the collection of all x for which f(x) is negative, and let Z be the collection of all x for which f(x) is zero. The natural domain of (gof)(x) is PUN. Explain. C. The natural domain of (ho f) (x) is PUZ. Explain.
Let f: R → R be a function, let function g: (-∞, 0) U (0, ∞) → R be defined by g(x) = 1, and let function h: [0, ∞) → R be defined by h(x)=√x. A. In terms of f(x), find an expression for the functions (gof)(x), (hof)(x), (gohof)(x), (hogof)(x). B. Henceforth, let P be the collection of all x for which f(x) is positive, let N be the collection of all x for which f(x) is negative, and let Z be the collection of all x for which f(x) is zero. The natural domain of (gof)(x) is PUN. Explain. C. The natural domain of (ho f) (x) is PUZ. Explain.
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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Transcribed Image Text:**Problem Statement:**
1. Let \( f : \mathbb{R} \to \mathbb{R} \) be a function, let function \( g: (-\infty, 0) \cup (0, \infty) \to \mathbb{R} \) be defined by \( g(x) = \frac{1}{x} \), and let function \( h: [0, \infty) \to \mathbb{R} \) be defined by \( h(x) = \sqrt{x} \).
A. In terms of \( f(x) \), find an expression for the functions \( (g \circ f)(x) \), \( (h \circ f)(x) \), \( (g \circ h \circ f)(x) \), \( (h \circ g \circ f)(x) \).
B. Henceforth, let \( P \) be the collection of all \( x \) for which \( f(x) \) is positive, let \( N \) be the collection of all \( x \) for which \( f(x) \) is negative, and let \( Z \) be the collection of all \( x \) for which \( f(x) \) is zero. The natural domain of \( (g \circ f)(x) \) is \( P \cup N \). Explain.
C. The natural domain of \( (h \circ f)(x) \) is \( P \cup Z \). Explain.
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