# The functions f o g , g o f , f o f , g o g and their domains for f ( x ) = 2 x and g ( x ) = x x + 2 .

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.6, Problem 44E
To determine

## To find:The functions fog , gof , fof , gog and their domains for f(x)=2x and g(x)=xx+2 .

Expert Solution

The function fog is equal to 2(x+2)x with domain (,0)(0,) , gof is equal to 11+x with domain set of all the real numbers except x=2 and x=1 , fof is equal to x with domain set of all the real numbers except x=0 , gog is equal to (gog)(x)=x3x+4 with domain set of real numbers except x=2 and x=43 . for f(x)=2x and g(x)=xx+2 .

### Explanation of Solution

Given information:

The functions are f(x)=2x and g(x)=xx+2 .

Calculation:

Simplify the function fog .

(fog)(x)=f(g(x))=f(xx+2)=2xx+2=2(x+2)x

The function f(x)=2x is defined for all the real numbers of x except x=0 and the function (fog)(x)=2(x+2)x is defined only if x0 . Combine both the conditions.

So, the domain of the function (fog)(x)=2(x+2)x is equal to (,0)(0,) .

Simplify the function gof .

(gof)(x)=g(f(x))=g(2x)=2x2x+2

Further simplify,

(gof)(x)=2x2x+2=2x2+2xx=11+x

The function g(x)=xx+2 is defined for all the real values of x except x=2 and the function (gof)(x)=11+x is defined only if 1+x0 i.e. x1 . Combine both the conditions.

So, the domain of the function (gof)(x)=11+x is set of all the real numbers except x=2 and x=1 .

Simplify the function fof .

(fof)(x)=f(f(x))=f(2x)=22x=x

The function f(x)=2x is defined for all the real numbers of x except x=0 and the function (fof)(x)=x is defined for all real values of x . Combine both the conditions.

So, the domain of the function (fof)(x)=x is set of all the real numbers except x=0 .

Simplify the function gog .

(gog)(x)=g(g(x))=g(xx+2)=xx+2xx+2+2

Further simplify,

(gog)(x)=xx+2xx+2+2=xx+2x+2(x+2)x+2=xx+2x+4=x3x+4

The function g(x)=xx+2 is defined for all the real values of x except x=2 and the function (gog)(x)=x3x+4 is defined only if 3x+40 i.e. x43 .

So, the domain of the function (gog)(x)=x3x+4 is set of real numbers except x=2 and x=43 .

Therefore, the function fog is equal to 2(x+2)x with domain (,0)(0,) , gof is equal to 11+x with domain set of all the real numbers except x=2 and x=1 , fof is equal to x with domain set of all the real numbers except x=0 , gog is equal to (gog)(x)=x3x+4 with domain set of real numbers except x=2 and x=43 . for f(x)=2x and g(x)=xx+2 .

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