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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
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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**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.
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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