BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 1.3, Problem 42E
To determine

To express: The function F(x)=cos2x in the form of (fg)(x) .

Expert Solution

Answer to Problem 42E

The function F(x)=cos2x can be expressed as F(x)=f(g(x)), where  f(x)=x2 and g(x)=cosx .

Explanation of Solution

The given composite function is F(x)=cos2x .

Reframe the given function as F(x)=(cosx)2 .

Name the expression cosx as g(x)=cosx .

Thus, F(x)=(g(x))2

It can be expressed as F(x)=f(g(x)) as F(x) is a composition of f and g.

Replace g(x) by x to obtain the expression for f(x) .

Therefore, f(x)=x2 .

Hence, it can be concluded that F(x)=cos2x can be expressed as F(x)=f(g(x)), where  f(x)=x2 and g(x)=cosx .

Verification:

The composite function (fg)(x) is defined as follows.

(fg)(x)=f(g(x))

Substitute cosx for x in f(x) .

f(cosx)=(cosx)2=cos2x=F(x)

Thus, it is verified that F(x)=cos2x can be expressed as F(x)=f(g(x)), where  f(x)=x2 and g(x)=cosx .

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