# The Function f such that f ∘ g = h . And using same sort of reasoning to find a function g such that f ∘ g = h . ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.6, Problem 69E
To determine

## To Find : The Function f such that f∘g=h . And using same sort of reasoning to find a function g such that f∘g=h .

Expert Solution

The Answer is, in part 1, f(x)=x2+6 and in part 2, g(x)=x2+x1 .

### Explanation of Solution

Given information : In 1st part, functions g(x)=2x+1 and h(x)=4x2+4x+7 is given for fg=h .

In 2nd part, functions f(x)=3x+5 and h(x)=3x2+3x+2 is given for fg=h .

Formula used : The concept of Unrecognized Combining Functions is used.

Calculation :

In part 1 g(x)=2x+1 and h(x)=4x2+4x+7

fg=hf(g(x)=h(x)f(2x+1)=4x2+4x+7

Now, transform h(x) in form of g(x)

h(x)=4x2+4x+7h(x)=(2x+1)2+6

On Comparing,

f(2x+1)= (2x+1) 2 +6f(x)=x2+6

In part 2 f(x)=3x+5 and h(x)=3x2+3x+2

fg=hf(g(x))=h(x)3g(x)+5=3x2+3x+23g(x)=3x2+3x3

Divide both sides by 3.

g(x)=x2+x1

Thus, answer is, in part 1 f(x)=x2+6 and in part 2, g(x)=x2+x1 .

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