Problem 9.2.13. Prove Theorem 9.2.12 Theorem 9.2.12. Suppose f is continuous at a and g is continuous at f(a). Then g of is continuous at a. (Note that (go f) (x) = g(f(x)),) in-context (a) Using the definition of continuity. (b) Using Theorem 9.2.1.
Problem 9.2.13. Prove Theorem 9.2.12 Theorem 9.2.12. Suppose f is continuous at a and g is continuous at f(a). Then g of is continuous at a. (Note that (go f) (x) = g(f(x)),) in-context (a) Using the definition of continuity. (b) Using Theorem 9.2.1.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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Question
100%
Analytical math
![12:41 M
A personal.psu.edu/ecb!
+
4
(Note that (g o f)(x) = g(f(x)).)
in-context
(a) Using the definition of continuity.
(b) Using Theorem 9.2.1.
Theorem 9.2.1. The function f is
continuous at a if and only if f satisfies
the following property:
sequences (xn), if lim xn = a then lir
A
in-context
The above theorems allow us to build
continuous functions from other
continuous functions. For example,
knowing that f(x) = x and g(x) = c are
continuous, we can conclude that any
polynomial,
p(æ)
n-1
anx" + an-1x"-
+ a1x + ao
is continuous as well. We also know that
functions such as f(x) = sin (e") are
continuous without having to rely on the
definition.
II
II](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0335de1d-d88b-4764-a43c-e2195c6bbbda%2Fa19f6a8a-a39b-489a-952a-e25ab3d9329b%2Ftk6jkrb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:12:41 M
A personal.psu.edu/ecb!
+
4
(Note that (g o f)(x) = g(f(x)).)
in-context
(a) Using the definition of continuity.
(b) Using Theorem 9.2.1.
Theorem 9.2.1. The function f is
continuous at a if and only if f satisfies
the following property:
sequences (xn), if lim xn = a then lir
A
in-context
The above theorems allow us to build
continuous functions from other
continuous functions. For example,
knowing that f(x) = x and g(x) = c are
continuous, we can conclude that any
polynomial,
p(æ)
n-1
anx" + an-1x"-
+ a1x + ao
is continuous as well. We also know that
functions such as f(x) = sin (e") are
continuous without having to rely on the
definition.
II
II
![12:41
Problem 9.2.13. Prove Theorem 9.2.12
Theorem 9.2.12. Suppose f is
continuous at a and g is continuous at
f(a). Then go f is continuous at a.
(Note that (go
f)(x) = g(f(x))
in-context
(a) Using the definition of continuity.
(b) Using Theorem 9.2.1.
Theorem 9.2.1. The function f is
continuous at a if and only if f satisfies
the following property:
V sequences (xn), if lim xn = a then lir
n 0
in-context
The above theorems allow us to build
continuous functions from other
continuous functions. For example,
knowing that f(x)
continuous, we can conclude that any
polynomial,
= x and g(x) = c are
p(x) = a,a"
+ an-1x"
n-1
+
+ ajx + ao
II
II](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0335de1d-d88b-4764-a43c-e2195c6bbbda%2Fa19f6a8a-a39b-489a-952a-e25ab3d9329b%2Fvfiow5g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:12:41
Problem 9.2.13. Prove Theorem 9.2.12
Theorem 9.2.12. Suppose f is
continuous at a and g is continuous at
f(a). Then go f is continuous at a.
(Note that (go
f)(x) = g(f(x))
in-context
(a) Using the definition of continuity.
(b) Using Theorem 9.2.1.
Theorem 9.2.1. The function f is
continuous at a if and only if f satisfies
the following property:
V sequences (xn), if lim xn = a then lir
n 0
in-context
The above theorems allow us to build
continuous functions from other
continuous functions. For example,
knowing that f(x)
continuous, we can conclude that any
polynomial,
= x and g(x) = c are
p(x) = a,a"
+ an-1x"
n-1
+
+ ajx + ao
II
II
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