# 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, Problem 19RE

(a)

To determine

Expert Solution

## Answer to Problem 19RE

Solution:

The function (fg)(x)=ln(x29) and its domain is (,3)(3,).

### Explanation of Solution

Given:

The functions are f(x)=lnx,g(x)=x29.

Calculation:

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

(fg)(x)=f(g(x))=f(x29)

Substitute x29 for x in f(x),

f(x29)=ln(x29)

Thus, the composite function (fg)(x)=ln(x29).

Note that the function f(x) is defined on all positive real numbers and hence the domain of f(x) is (0,).

The function g(x) is defined on real numbers and hence the domain of the function g(x) is (,).

To find the domain of (fg)(x)=ln(x29) set the expression x29>0 and solve.

x29>0x2>9|x|>3

Thus, the required domain is (,3)(3,).

(b)

To determine

Expert Solution

## Answer to Problem 19RE

Solution:

The function (gf)(x)=(lnx)29 and its domain is (0,).

### Explanation of Solution

Given:

The functions are f(x)=lnx,g(x)=x29.

Calculation:

The composite function (gf)(x) is defined as follows.

(gf)(x)=g(f(x))=g(lnx)

Substitute lnx for x in g(x).

g(lnx)=(lnx)29

Thus, the composite function (gf)(x)=(lnx)29.

Since the function (gf)(x) is defined on positive real numbers, the domain of the composite function (gf)(x) is (0,).

(c)

To determine

Expert Solution

## Answer to Problem 19RE

Solution:

The composite function (ff)(x)=ln(lnx) and its domain is (,).

### Explanation of Solution

Given:

The function is f(x)=lnx.

The composite function (ff)(x) is defined as follows.

(ff)(x)=f(f(x))=f(lnx)

Substitute lnx for x in f(x).

f(lnx)=ln(lnx)

Thus, the composite function (ff)(x)=ln(lnx).

Since, the function f(x) is defined on positive real numbers, the domain of the function f(x) is (0,).

Find the domain of the function (ff)(x)=ln(lnx) as follows.

Set the expression, lnx>0.

Take exponential on both sides,

elnx>e0x>1

Therefore, the domain of the composite function (ff)(x) is (1,).

(d)

To determine

Expert Solution

## Answer to Problem 19RE

Solution:

The composite function (gg)(x)=x418x2+72 and its domain is (,).

### Explanation of Solution

Given:

The function is g(x)=x29.

Calculation:

The composite function (gg)(x) is defined as follows.

(gg)(x)=g(g(x))=g(x29)

Substitute x29 for x in g(x),

g(x29)=(x29)29=x42x2(9)+819=x418x2+72

Thus, the composite function (gg)(x)=x418x2+72.

Since the function (gg)(x) is defined on all real numbers, the domain of the composite function (gg)(x) is (,).

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