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 41E
To determine

To express: The function F(x)=(2x+x2)4 in the form of (fg)(x) .

Expert Solution

Answer to Problem 41E

The function F(x)=(2x+x2)4 can be expressed as F(x)=f(g(x)), where  f(x)=x4 and g(x)=2x+x2 .

Explanation of Solution

The given composite function is F(x)=(2x+x2)4 .

Name the expression 2x+x2 as g(x)=2x+x2 .

Thus, F(x)=(g(x))4 .

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)=x4 .

Hence, it can be concluded that F(x)=(2x+x2)4 can be expressed as F(x)=f(g(x)), where  f(x)=x4 and g(x)=2x+x2 .

Verification:

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

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

Substitute 2x+x2 for x in f(x) .

f(2x+x2)=(2x+x2)4=F(x)

Thus, it is verified that F(x)=(2x+x2)4 can be expressed as F(x)=f(g(x)), where  f(x)=x4 and g(x)=2x+x2 .

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