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 47E
To determine

To express: The function R(x)=x1 in the form of (fgh)(x) .

Expert Solution

Answer to Problem 47E

Solution:

The function R(x)=x1 can be expressed as R(x)=f(g(h(x))), where f(x)=x,g(x)=x1 and  h(x)=x .

Explanation of Solution

Given:

The composite function is R(x)=x1 .

Calculation:

Name the expression x as h(x)=x .

Thus, R(x)=h(x)1 .

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

Therefore, R(x)=x1 .

Name the expression x1 as g(x)=x1 .

Thus, R(x)=g(x)

Similarly, replace g(x) by x to obtain the expression for f(x) .

Therefore, f(x)=x .

Hence, it can be concluded that R(x)=x1 can be expressed as R(x)=f(g(h(x))), where f(x)=x,g(x)=x1 and  h(x)=x .

Verification:

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

(fgh)(x)=f(gh)(x)

First find the composite function of (gh)(x) .

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

(gh)(x)=g(h(x))

Substitute x for x in g(x) .

g(x)=x1

Thus, the composite function (gh)(x)=x1 .

Substitute x1 for x in f(x) and find the composition function (fgh)(x) .

f(x1)=x1=R(x)

Therefore, the composite function (fgh)(x)=x1 .

Thus, it is verified that R(x)=x1 can be expressed as R(x)=f(g(h(x))), where f(x)=x,g(x)=x1 and  h(x)=x .

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