# To express: The function F ( x ) = 1 x + x in the composition of three functions. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

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

#### Solutions

Chapter 1, Problem 20RE
To determine

## To express: The function F(x)=1x+x in the composition of three functions.

Expert Solution

Solution:

The function F(x)=1x+x can be expressed as F(x)=f(g(h(x))), where f(x)=1x,g(x)=x2+x and  h(x)=x.

### Explanation of Solution

The composite function is F(x)=1x+x.

Express the function as, F(x)=1(x)2+x

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

Thus, F(x)=1(h(x))2+h(x).

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

Therefore, F(x)=1(x)2+x.

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

Thus, F(x)=1g(x).

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

Therefore, f(x)=1x.

Hence, it can be concluded that F(x)=1x+x can be expressed as F(x)=f(g(h(x))), where f(x)=1x,g(x)=x2+x 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)=(x)2+x=x2+x

Thus, the composite function (gh)(x)=x2+x.

Substitute x2+x for x in f(x) and find the composition function (fgh)(x).

f(x+x)=1x+x=F(x)

Therefore, the composite function (fgh)(t)=1x+x.

Thus, it is verified that F(x)=1x+x can be expressed as F(x)=f(g(h(x))), where f(x)=1x,g(x)=x2+x and  h(x)=x.

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