# 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 32E

a.

To determine

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

Expert Solution

(fg)(x)=x2+3x+2;domain .

### Explanation of Solution

Given:

The given function is f(x)=x2,g(x)=x2+3x+4 .

Calculation:

As the given function is f(x)=x2,g(x)=x2+3x+4 .

(fg)(x)=f(g(x))=(x2+3x+4)2{as f(x)=x2}=x2+3x+2

Note that the domain of the function is a polynomial function thus, domain .

Hence,the range and domain are (fg)(x)=x2+3x+2;domain

b.

To determine

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

Expert Solution

(gf)(x)=x23x+2;domain .

### Explanation of Solution

Given:

The given function is f(x)=x2,g(x)=x2+3x+4 .

Calculation:

As the given function is f(x)=x2,g(x)=x2+3x+4 .

(gf)(x)=f(g(x))=(x2)2+3(x2)+4{as g(x)=x2+3x+4}=x26x+4+3x6+4=x23x+2

Note that the domain of the function is a polynomial function thus, domain .

Hence, the range and domain are (gf)(x)=x23x+2;domain

c.

To determine

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

Expert Solution

(ff)(x)=x4;domain .

### Explanation of Solution

Given:

The given function is f(x)=x2,g(x)=x2+3x+4 .

Calculation:

As the given function is f(x)=x2,g(x)=x2+3x+4 .

(ff)(x)=f(f(x))=(x2)2{as f(x)=x2}=x4

Note that the domain of the function is a polynomial function thus, domain .

Hence, the range and domain are (ff)(x)=x4;domain

d.

To determine

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

Expert Solution

(gg)(x)=x4+6x3+20x2+33x+32;domain .

### Explanation of Solution

Given:

The given function is f(x)=x2,g(x)=x2+3x+4 .

Calculation:

As the given function is f(x)=x2,g(x)=x2+3x+4 .

(gf)(x)=f(g(x))=(x2+3x+4)2+3(x2+3x+4)+4{as g(x)=x2+3x+4}=x4+(3x)2+(4)2+2x2×(3x)+2(3x)(4)+2(4)x2+3x2+9x+12+4=x4+9x2+16+6x3+24x+8x2+3x2+9x+12+4=x4+6x3+20x2+33x+32

Note that the domain of the function is a polynomial function thus, domain .

Hence, the range and domain are (gg)(x)=x4+6x3+20x2+33x+32;domain

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