# 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 36E

a.

To determine

Expert Solution

## Answer to Problem 36E

(fg)(x)=sin2x1+2sin2x;{nπ±(1)n(3π4)} .

### Explanation of Solution

Given:

The given function is f(x)=11+x,g(x)=sin2x .

Calculation:

As the given function is f(x)=x1+x,g(x)=sin2x .

(fg)(x)=f(g(x))=sin2x1+2sin2x{as f(x)=x1+x}

Therefore, the domain of the function is all value of x expect sin2x=1 .

Therefore, the general solution for xis x=nπ±(1)n(3π4) .

Therefore, the domain is {nπ±(1)n(3π4)}

Hence,the range and domain are (fg)(x)=sin2x1+2sin2x;{nπ±(1)n(3π4)}

b.

To determine

Expert Solution

## Answer to Problem 36E

(gf)(x)=sin(2x1+2x);{1} .

### Explanation of Solution

Given:

The given function is f(x)=11+x,g(x)=sin2x .

Calculation:

As the given function is f(x)=x1+x,g(x)=sin2x .

(fg)(x)=f(g(x))=sin(2x1+2x){as g(x)=sin2x}

Therefore, the domain is {1} .

Hence, the range and domain are (gf)(x)=sin(2x1+2x);{1}

c.

To determine

Expert Solution

## Answer to Problem 36E

(ff)(x)=x1+2x;{1,12} .

### Explanation of Solution

Given:

The given function is f(x)=11+x,g(x)=sin2x .

Calculation:

As the given function is f(x)=x1+x,g(x)=sin2x .

(fg)(x)=f(g(x))=x1+x1+x1+x{as f(x)=x1+x}=x1+x1+x+x1+x=x1+2x

Since, g(x) is not defined at x=1,12

Therefore, the domain is {1,12}

Hence, the range and domain are (ff)(x)=x1+2x;{1,12}

d.

To determine

Expert Solution

## Answer to Problem 36E

(gg)(x)=sin(sin2x); domain is  .

### Explanation of Solution

Given:

The given function is f(x)=11+x,g(x)=sin2x .

Calculation:

As the given function is f(x)=x1+x,g(x)=sin2x .

(gg)(x)=(g(x))=sin(sin2x){as g(x)=sin2x}

Therefore, the domain of the function is .

Hence, the range and domain are (gg)(x)=sin(sin2x); domain is

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