### Problem Statement: **6. Find \( g \circ f \)**For each part below, find the composition of functions \( g \circ f \) where \( g \circ f(x) = g(f(x)) \).---**(a)** \( f: \mathbb{Z} \to \mathbb{N}, \quad f(n) = n^2 + 1; \) \( g: \mathbb{N} \to \mathbb{Q}, \quad g(n) = \frac{1}{n}. \)**(b)** \( f: \mathbb{R} \to (0, 1), \quad f(x) = \frac{1}{(x^2 + 1)}; \) \( g: (0, 1) \to (0, 1), \quad g(x) = 1 - x. \)**(c)** \( f: \mathbb{Q} - \{2\} \to \mathbb{Q}^*, \quad f(x) = \frac{1}{(x - 2)}; \) \( g: \mathbb{Q}^* \to \mathbb{Q}^*, \quad g(x) = \frac{1}{x}. \)**(d)** \( f: \mathbb{R} \to [1, \infty), \quad f(x) = x^2 + 1; \) \( g: [1, \infty) \to [0, \infty), \quad g(x) = \sqrt{x - 1}. \)**(e)** \( f: \mathbb{Q} - \left\{\frac{10}{3}\right\} \to \mathbb{Q} - \{3\}, \quad f(x) = 3x - 7; \) \( g: \mathbb{Q} - \{3\} \to \mathbb{Q} - \{2\}, \quad g(x) = \frac{2x}{(x - 3)}. \)

Answered: Image /qna-images/answer/8b4d4eae-855a-455e-9cd5-6806d3d96ba2.jpg

Find g of (a) f:Z→ N, (b) f:R → (0,1), f(x)=1/(x² +1); f(n) = n² + 1; g:N → Q, g(n) = g: (0, 1) → (0, 1), g(x) = 1 – x. (c) f:Q – {2} → Q*, f(x) = 1/(x – 2); g: Q* → Q*, g(x) = 1/x. g:[1, 00) → [0, 0) g(x) = Vr – I. g:Q – {3} → Q - {2}, (d) f:R → [1, 0), f(x) = x² + 1; (e) f:Q – {10/3} → Q – {3}, ƒ(x) = 3x – 7; 2a/(x – 3).

Find g of (a) f:Z→ N, (b) f:R → (0,1), f(x)=1/(x² +1); f(n) = n² + 1; g:N → Q, g(n) = g: (0, 1) → (0, 1), g(x) = 1 – x. (c) f:Q – {2} → Q, f(x) = 1/(x – 2); g: Q → Q*, g(x) = 1/x. g:[1, 00) → [0, 0) g(x) = Vr – I. g:Q – {3} → Q - {2}, (d) f:R → [1, 0), f(x) = x² + 1; (e) f:Q – {10/3} → Q – {3}, ƒ(x) = 3x – 7; 2a/(x – 3).

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Representation of Polynomial

It is necessary for the polynomial expression that each expression consists of two parts:

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### Problem Statement:

**6. Find \( g \circ f \)**

For each part below, find the composition of functions \( g \circ f \) where \( g \circ f(x) = g(f(x)) \).

---

**(a)** \( f: \mathbb{Z} \to \mathbb{N}, \quad f(n) = n^2 + 1; \)
\( g: \mathbb{N} \to \mathbb{Q}, \quad g(n) = \frac{1}{n}. \)

**(b)** \( f: \mathbb{R} \to (0, 1), \quad f(x) = \frac{1}{(x^2 + 1)}; \)
\( g: (0, 1) \to (0, 1), \quad g(x) = 1 - x. \)

**(c)** \( f: \mathbb{Q} - \{2\} \to \mathbb{Q}^*, \quad f(x) = \frac{1}{(x - 2)}; \)
\( g: \mathbb{Q}^* \to \mathbb{Q}^*, \quad g(x) = \frac{1}{x}. \)

**(d)** \( f: \mathbb{R} \to [1, \infty), \quad f(x) = x^2 + 1; \)
\( g: [1, \infty) \to [0, \infty), \quad g(x) = \sqrt{x - 1}. \)

**(e)** \( f: \mathbb{Q} - \left\{\frac{10}{3}\right\} \to \mathbb{Q} - \{3\}, \quad f(x) = 3x - 7; \)
\( g: \mathbb{Q} - \{3\} \to \mathbb{Q} - \{2\}, \quad g(x) = \frac{2x}{(x - 3)}. \)

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