EXAMPLE 9 Video Example 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)=x² and f(x) = X We know that g is continuous on R since it is a polynomial and f is also continuous everywhere. Thus, fog 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. The expression In(1 + cos(x)) is defined when 1 + cos(x) > . So it is undefined when cos(x) = Thus, F has discontinuities when x is an odd multiple of it and is continuous on the intervals between these values. (See the figure below.) 10 5 mdr. T -10 L -10 10 X G , and this happens when x ±, ±3m,....
EXAMPLE 9 Video Example 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)=x² and f(x) = X We know that g is continuous on R since it is a polynomial and f is also continuous everywhere. Thus, fog 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. The expression In(1 + cos(x)) is defined when 1 + cos(x) > . So it is undefined when cos(x) = Thus, F has discontinuities when x is an odd multiple of it and is continuous on the intervals between these values. (See the figure below.) 10 5 mdr. T -10 L -10 10 X G , and this happens when x ±, ±3m,....
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:EXAMPLE 9
Video Example
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)=x² and f(x) =
X
We know that g is continuous on R since it is a polynomial and f is also continuous everywhere. Thus, fog 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.
The expression In(1 + cos(x)) is defined when 1 + cos(x) >
. So it is undefined when cos(x) =
Thus, F has discontinuities when x is an odd multiple of it and is continuous on the intervals between these values. (See the figure below.)
10
5
mdr.
T
-10 L
-10
10
X
G
, and this happens when x ±, ±3m,....
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