# To check: Whether f ∘ g is linear if f and g are linear; find the slope of the graph f ∘ g if it is linear. ### 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 59E
To determine

## To check: Whether f∘g is linear if f and g are linear; find the slope of the graph f∘g if it is linear.

Expert Solution

The composition function fg is a linear function of the slope m1m2 .

### Explanation of Solution

Given:

The linear functions are f(x)=m1x+b1,g(x)=m2x+b2 .

Calculation:

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

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

Substitute m2x+b2 for x in f(x) and find the composition function (fg)(x) .

f(g(x))=m1(m2x+b2)+b1=m1m2x+m1b2+b1

Observe that (fg)(x) in the form of the linear function y=mx+c where m is slope and c is constant.

Thus, it can be concluded that the composite function (fg)(x) is a linear function with the slope m1m2 .

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