# 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 30E

(a)

To determine

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

Expert Solution

The sum of the functions f(x)+g(x)=3x+x21 and its domain is (,1][1,3].

### Explanation of Solution

Given:

The functions are f(x)=3x,g(x)=x21.

Result used:

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

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

Calculation:

Substitute the value of f(x) and g(x) in (f+g)(x)=f(x)+g(x).

f(x)+g(x)=3x+x21

As the radicand cannot take negative values, 3x>0.

On simplification,

3x>0x<3

Also x21>0.

Simplify the above inequality as follows.

x21>0x2>1x>±1

So the value of x lies in the interval x<3,x>±1.

Hence, the domain of 3x is (,3] and the domain of x21 is (,1][1,).

So the combined domain is (,1][1,3].

Therefore, the domain of f(x)+g(x)=3x+x21 is (,1][1,3].

(b)

To determine

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

Expert Solution

The function fg is f(x)g(x)=3xx21 and its domain is (,1][1,3].

### Explanation of Solution

Result used:

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

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

Calculation:

Substitute the value of f(x) and g(x) in (fg)(x)=f(x)g(x).

f(x)g(x)=3xx21

From part (a), the domain of the function f+g is (,1][1,3], which is also the domain of fg.

Therefore, the value of the given function is f(x)g(x)=3xx21 whose domain is (,1][1,3].

(c)

To determine

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

Expert Solution

The product function f(x)g(x)=(3x)(x21) and its domain is (,1][1,3].

### Explanation of Solution

Result used:

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

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

Calculation:

Substitute the value of f(x) and g(x) in (fg)(x)=f(x)g(x).

f(x)g(x)=(3x)(x21).

From part (a), the domain of the function f+g is (,1][1,3], which is also the domain of fg.

Therefore, the value of the given function is f(x)g(x)=(3x)(x21) whose domain is (,1][1,3].

(d)

To determine

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

Expert Solution

The quotient function is f(x)g(x)=(3x)(x21) and its domain is (,1)(1,3].

### Explanation of Solution

Result used:

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

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

Calculation:

Substitute the value of f(x) and g(x) in (fg)(x)=f(x)g(x), f(x)g(x)=(3x)(x21).

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

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

Simplify the equation x21=0.

x21=0x2=1x=±1

So, the numbers −1 and 1 are excluded from the domain.

From part (a), the domain of the function f+g is (,1][1,3].

Remove −1 and 1 from (,1][1,3], (,1)(1,3].

Therefore, the domain of the function is (,1)(1,3].

Therefore, the value of the given function is f(x)g(x)=(3x)(x21) whose domain is (,1)(1,3].

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