# The functions f o g , g o f , f o f , g o g and their domains for f ( x ) = 1 x and g ( x ) = x 2 − 4 x . ### 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 42E
To determine

## To find:The functions fog , gof , fof , gog and their domains for f(x)=1x and g(x)=x2−4x .

Expert Solution

The function fog is equal to 1x24x with domain (,0)(4,) , gof is equal to 1x4x with domain (0,) , fof is equal to x14 with domain [0,) , gog is equal to x48x3+12x2+16x with domain (,) for f(x)=1x and g(x)=x24x .

### Explanation of Solution

Given information:

The functions are f(x)=1x and g(x)=x24x .

Calculation:

Simplify the function fog .

(fog)(x)=f(g(x))=f(x24x)=1x24x

The function g(x)=x24x is defined for all the real values of x and the function (fog)(x)=1x24x is defined only if x24x>0 it is possible only if x>4 or x<0 .

So, the domain of the function (fog)(x)=1x24x is equal to (,0)(4,) .

Simplify the function gof .

(gof)(x)=g(f(x))=g(1x)=(1x)24×1x=1x4x

The function f(x)=1x is defined for all the values of x0 and the function (gof)(x)=1x4x is defined only if x>0 . Combine both the conditions.

So, the domain of the function (gof)(x)=1x4x is equal to (0,) .

Simplify the function fof .

(fof)(x)=f(f(x))=f(1x)=11x=x14

The function f(x)=1x is defined for all the real values of x0 and the function (fof)(x)=x14 is defined only if x0 . Combine both the conditions.

So, the domain of the function (fof)(x)=x14 is equal to [0,) .

Simplify the function gog .

(gog)(x)=g(g(x))=g(x24x)=(x24x)24(x24x)

Further simplify,

(gog)(x)=(x24x)24(x24x)=x4+16x28x34x2+16x=x48x3+12x2+16x

The function g(x)=x24x is defined for all the real values of x and the function (gog)(x)=x48x3+12x2+16x is defined for all the real values of x . Combine both the conditions.

So, the domain of the function (gog)(x)=x48x3+12x2+16x is equal to (,) .

Therefore, the function fog is equal to 1x24x with domain (,0)(4,) , gof is equal to 1x4x with domain (0,) , fof is equal to x14 with domain [0,) , gog is equal to x48x3+12x2+16x with domain (,) for f(x)=1x and g(x)=x24x .

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