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EXAMPI Where are the following functions continuous? (a) h(x) = sin(x8) (b) F(x) = In(1 + cos(x)) SOLUTION (a) We have h(x) = f(g(x)), where g(x) = x and f(x) = Now g is continuous on R since it is a polynomial and f is also continuous everywhere. Thus f o g is continuou on R by this theorem. (b) We know from this theorem that f(x) = In(x) is continuous and g(x) = 1 + cos(x) is continuous (because both y = 1 and y = cos(x) are continuous). Therefore, by this theorem, F(x) = f(g(x)) is continuous wherever it is defined. Now In(1 + cos(x)) is defined when 1 + cos(x) > . So it is undefined when cos(x) = , and this happens when x = ±, ±37, ..... Thus F has discontinuities when x is an odd multiple of and is continuous on the intervals between these values (see the figure).

EXAMPI Where are the following functions continuous? (a) h(x) = sin(x8) (b) F(x) = In(1 + cos(x)) SOLUTION (a) We have h(x) = f(g(x)), where g(x) = x and f(x) = Now g is continuous on R since it is a polynomial and f is also continuous everywhere. Thus f o g is continuou on R by this theorem. (b) We know from this theorem that f(x) = In(x) is continuous and g(x) = 1 + cos(x) is continuous (because both y = 1 and y = cos(x) are continuous). Therefore, by this theorem, F(x) = f(g(x)) is continuous wherever it is defined. Now In(1 + cos(x)) is defined when 1 + cos(x) > . So it is undefined when cos(x) = , and this happens when x = ±, ±37, ..... Thus F has discontinuities when x is an odd multiple of and is continuous on the intervals between these values (see the figure).

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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-10 -5 y 10 5 -5 -10 5 10 X
Transcribed Image Text:-10 -5 y 10 5 -5 -10 5 10 X
EXAMPLE 9 Where are the following functions continuous? (a) h(x) = sin(x³) (b) F(x) = In(1 + cos(x)) SOLUTION (a) We have h(x) = f(g(x)), where g(x) = x and f(x) = Now g is continuous on R since it is a polynomial and f is also continuous everywhere. Thus fo g is continuous on R by this theorem. (b) We know from this theorem that f(x) = In(x) is continuous and g(x) = 1 + cos(x) is continuous (because both y = 1 and y = cos(x) are continuous). Therefore, by this theorem, F(x) = f(g(x)) is continuous wherever it is defined. Now In(1 + cos(x)) is defined when 1 + cos(x) > . So it is undefined when cos(x) = , and this happens when x = ±, ±3, .... Thus F has discontinuities when x is an odd multiple of and is continuous on the intervals between these values (see the figure).
Transcribed Image Text:EXAMPLE 9 Where are the following functions continuous? (a) h(x) = sin(x³) (b) F(x) = In(1 + cos(x)) SOLUTION (a) We have h(x) = f(g(x)), where g(x) = x and f(x) = Now g is continuous on R since it is a polynomial and f is also continuous everywhere. Thus fo g is continuous on R by this theorem. (b) We know from this theorem that f(x) = In(x) is continuous and g(x) = 1 + cos(x) is continuous (because both y = 1 and y = cos(x) are continuous). Therefore, by this theorem, F(x) = f(g(x)) is continuous wherever it is defined. Now In(1 + cos(x)) is defined when 1 + cos(x) > . So it is undefined when cos(x) = , and this happens when x = ±, ±3, .... Thus F has discontinuities when x is an odd multiple of and is continuous on the intervals between these values (see the figure).
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