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Part. f. True or False: g(x) = In(x-1) is continuous for x > 1. 4 x +22

Part. f. True or False: g(x) = In(x-1) is continuous for x > 1. 4 x +22

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
### Part f. **True or False**: \( g(x) = \frac{\ln(x-1)}{x^4 + 22} \) is continuous for \( x > 1 \). **Explanation**: - The function \( g(x) = \frac{\ln(x-1)}{x^4 + 22} \) consists of a natural logarithm function in the numerator and a polynomial function in the denominator. - The logarithmic function \( \ln(x-1) \) is defined and continuous for \( x > 1 \). - The polynomial function in the denominator, \( x^4 + 22 \), is always positive and continuous for all \( x \). - Since both the numerator and denominator are continuous for \( x > 1 \) and the denominator does not equal zero (ensuring no points of discontinuity), the function \( g(x) \) is continuous for \( x > 1 \). Hence, the statement is **True**.
Transcribed Image Text:### Part f. **True or False**: \( g(x) = \frac{\ln(x-1)}{x^4 + 22} \) is continuous for \( x > 1 \). **Explanation**: - The function \( g(x) = \frac{\ln(x-1)}{x^4 + 22} \) consists of a natural logarithm function in the numerator and a polynomial function in the denominator. - The logarithmic function \( \ln(x-1) \) is defined and continuous for \( x > 1 \). - The polynomial function in the denominator, \( x^4 + 22 \), is always positive and continuous for all \( x \). - Since both the numerator and denominator are continuous for \( x > 1 \) and the denominator does not equal zero (ensuring no points of discontinuity), the function \( g(x) \) is continuous for \( x > 1 \). Hence, the statement is **True**.
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