# The function f + g and its 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 29E

(a)

To determine

## To find: The function f+g and its domain.

Expert Solution

The sum of the functions (f+g)(x)=x3+5x21 and its domain is (,) .

### Explanation of Solution

Given:

The functions are f(x)=x3+2x2,g(x)=3x21 .

Calculation:

The sum of the functions (f+g)(x) is defined as follows.

(f+g)(x)=f(x)+g(x)

Substitute f(x) and g(x) ,

f(x)+g(x)=x3+2x2+3x21=x3+5x21

Notice that f(x)+g(x) is a polynomial function.

Since polynomial function is defined on real line, the domain of (f+g)(x) is a set of all real numbers.

Thus, (f+g)(x)=x3+5x21 whose domain is .

(b)

To determine

### To find: The function f−g and its domain.

Expert Solution

The subtraction of functions (fg)(x)=x3x2+1 and its domain is (,) .

### Explanation of Solution

Given:

The functions are f(x)=x3+2x2,g(x)=3x21 .

Calculation:

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

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

Substitute f(x) and g(x) ,

f(x)g(x)=x3+2x2(3x21)=x3+2x23x2+1=x3x2+1

Notice that (fg)(x) is a polynomial function.

Since polynomial function is defined on real line, the domain of (fg)(x) is a set of all real numbers.

Thus, (fg)(x)=x3x2+1 whose domain is .

(c)

To determine

### To find: The function fg and its domain.

Expert Solution

The product function (fg)(x)=3x5+6x4x32x2 and its domain is (,) .

### Explanation of Solution

Given:

The functions are f(x)=x3+2x2,g(x)=3x21 .

Calculation:

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

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

Substitute f(x) and g(x) ,

f(x)g(x)=(x3+2x2)(3x21)=(3x2)x3x3+2x2(3x2)2x2=3x5x3+6x42x2=3x5+6x4x32x2

Notice that (fg)(x) is a polynomial function.

Since polynomial function is defined on real line, the domain of (fg)(x) is a set of all real numbers.

Thus, (fg)(x)=3x5+6x4x32x2 whose domain is .

(d)

To determine

### To find: The function fg and its domain.

Expert Solution

The quotient function f(x)g(x)=x3+2x23x21 and its domain is x{±13} .

### Explanation of Solution

Given:

The functions are f(x)=x3+2x2,g(x)=3x21 .

Calculation:

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

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

Substitute f(x) and g(x) , f(x)g(x)=x3+2x23x21 .

Thus, the quotient function is defined when the denominator is not equal to 0.

Hence, the function is f(x)g(x)=x3+2x23x21 not defined when, 3x21=0 .

That is, x=±13 .

Therefore, the domain of the function is x{±13} .

Thus, f(x)g(x)=x3+2x23x21 whose domain is x{±13} .

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