Calculus, 8th Edition,stewart; page 44 problem 37

Question

Find the functions (a) f o g , (b) g o f , (c) f o f , and (d) g o g and their domains.

f (x) = x + 1/x,     g (x) = (x + 1)/(x + 2)

Answer 1

Step:1

The functions are  and .

 :

Rewrite the function.

All possible values of  is the domain of the function.

The denominator of the function should not be zero.

So .

The domain of  is .

 :

All possible values of  is the domain of the function.

The denominator of the function should not be zero.

So .

.

The domain of  is .

Step:2

(a)

Find .

 

.

The domain of the inside function  is .

All possible values of  is the domain of the function.

The denominator of the composite function should not be zero.

 and .

 and .

The domain of  is .

Step:3

(b)

Find .

 

.

The domain of the inside function  is .

All possible values of  is the domain of the function.

The denominator of the composite function should not be zero.

 

The domain of  is .

Comments

Step:4

(c)

Find .

 

.

The domain of the inside function  is .

All possible values of  is the domain of the function.

The denominator of the composite function should not be zero.

, 

 and 

Here  is not possible, so it can not be consider.

The domain of  is .

Step:5

(d)

Find .

 

.

The domain of the inside function  is .

All possible values of  is the domain of the function.

The denominator of the composite function should not be zero.

The domain of  is .

Solution:

(a) , and its domain is .

(b) , and its domain is .

(c) , and its domain is .

(d) , and its domain is .