# To express: The function H ( x ) = 2 + | x | 8 in the form of ( f ∘ g ∘ h ) ( x ) .

### 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 1.3, Problem 48E
To determine

## To express: The function H(x)=2+|x|8 in the form of (f∘g∘h)(x) .

Expert Solution

The function H(x)=2+|x|8 can be expressed as H(x)=f(g(h(x))), where f(x)=x8,g(x)=2+x 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, H(x)=2+h(x)8 .

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

Therefore, H(x)=2+x8 .

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

Thus, H(x)=g(x)8 .

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

Therefore, f(x)=x8 .

Hence, it can be concluded that H(x)=2+|x|8 can be expressed as H(x)=f(g(h(x))), where f(x)=x8,g(x)=2+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|)=2+|x|

Thus, the composite function (gh)(x)=2+|x| .

Substitute 2+|x| for x in f(x) and find the composition function (fgh)(x) .

f(2+|x|)=2+|x|8=H(x)

Therefore, the composite function (fgh)(x)=2+|x|8 .

Thus, it is verified that H(x)=2+|x|8 can be expressed as H(x)=f(g(h(x))), where f(x)=x8,g(x)=2+x and  h(x)=|x| .

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