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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) If f o ( x ) = 1 2 x and f n + 1 = f o f n  for  n =   0 ,   1 ,   2 ,   , find an expression for fn(x) and use mathematical induction to prove it.

(b) Graph f0, f1, f2, f3 on the same screen and describe the effects of repeated composition.

(a)

To determine

To find: The expression for f n ( x ) .

Explanation

Given:

f 0 ( x ) = 1 ( 2 x ) .

f n + 1 f 0 ( f n ( x ) )  for n = 0 , 1 , 2...

Calculation:

Base case: n = 1 .

To prove that the statement is true for n = 1 .

f n + 1 = f 0 ( f ( x ) 1 ) f 2 = f 0 ( f ( x ) 1 )

Therefore, the statement is true for n = 1 .

Induction hypothesis: n = k

Assuming that the claim is true for n = k .

That is, f k + 1 ( x ) = f 0 ( f k ( x ) )

Inductive step: n = k + 1

To prove the statement is true for n = k + 1 .

f 0 ( x ) = 1 ( 2 x )

Substitute n = 1 ,

f 1 ( x ) = f 0 ( 1 ( 2 x ) )

f 1 ( x ) = 1 2 ( 1 ( 2 x ) ) f 1 ( x ) = 2 x 3 2 x

Substitute n = 2 ,

f 2 ( x ) = 1 2 ( 2 x 3 2 x ) f 2 ( x ) = 3 2 x 6 4 x ( 2 x ) f 2 ( x ) = 3 2 x 4 3 x

Substitute

(b)

To determine

To graph: The function f 0 , f 1 , f 2 , f 3  .

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