# the given function f ∘ g and their domain.

### 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 31E

a.

To determine

## To find:the given function f∘g and their domain.

Expert Solution

(fg)(x)=4x2+4x;domain .

### Explanation of Solution

Given:

The given function is f(x)=x21,g(x)=2x+1 .

Calculation:

As the given function is f(x)=x21,g(x)=2x+1 .

The domain of the function is a real number, since both f and g are defined for all real numbers.

Now, compute the composite function directly.

(fg)(x)=f(g(x))=(2x+1)21{as f(x)=x21}=4x2+4x+11=4x2+4x

Note thatthe domain of the function is a real numberthus, domain .

Hence,the range and domain are (fg)(x)=4x2+4x;domain

b.

To determine

### To find: the given function g∘f and their domain.

Expert Solution

(gf)(x)=2x21;domain .

### Explanation of Solution

Given:

The given function is f(x)=x21,g(x)=2x+1 .

Calculation:

As the given function is f(x)=x21,g(x)=2x+1 .

The domain of the function is a real number, since both f and g are defined for all real numbers.

Now, compute the composite function directly.

(gf)(x)=g(f(x))=2(x21)+1{as g(x)=2x+1}=2x22+1=2x21

Note thatthe domain of the function is a real numberthus, domain .

Hence, the range and domain are (gf)(x)=2x21;domain

c.

To determine

### To find: the given function f∘f and their domain.

Expert Solution

(ff)(x)=x42x2;domain .

### Explanation of Solution

Given:

The given function is f(x)=x21,g(x)=2x+1 .

Calculation:

As the given function is f(x)=x21,g(x)=2x+1 .

The domain of the function is a real number, since both f and g are defined for all real numbers.

Now, compute the composite function directly.

(ff)(x)=g(f(x))=(x21)21{as f(x)=x21}=x42x2+11=x42x2

Note thatthe domain of the function is a real numberthus, domain .

Hence, the range and domain are (ff)(x)=x42x2;domain

d.

To determine

### To find: the given function g∘g and their domain.

Expert Solution

(gg)(x)=4x+3;domain .

### Explanation of Solution

Given:

The given function is f(x)=x21,g(x)=2x+1 .

Calculation:

As the given function is f(x)=x21,g(x)=2x+1 .

The domain of the function is a real number, since both f and g are defined for all real numbers.

Now, compute the composite function directly.

(gg)(x)=g(f(x))=2(2x+1)+1{as g(x)=2x+1}=4x+2+1=4x+3

Note thatthe domain of the function is a real numberthus, domain .

Hence, the range and domain are (gg)(x)=4x+3;domain

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