# To find the functions f o g , g o f , f o f , g o g and their domains of given f ( x ) = 3 x − 1 and g ( x ) = 2 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, Problem 77RE
To determine

## To find the functions fog , gof , fof , gog and their domains of given f(x)=3x−1 and g(x)=2x−x2

Expert Solution

The function fog(x)=3x2+6x1 and the domain of fog is (,) because the natural domain of any polynomial functions is <x< .

The function gof(x)=9x2+12x3 and the domain is (,) .

The function  fof(x)=9x4 and the domain is (,) .

The function gog(x)=x4+4x36x2+4x and the domain is (,) .

### Explanation of Solution

The given function are,

f(x)=3x1g(x)=2xx2

Using the definition of composite function,

fog(x)=f(g(x))=f(2xx2)=3(2xx2)1=6x3x21 fog(x)=3x2+6x1

Therefore, the domain of fog is (,) because the natural domain of any polynomial functions is <x< .

gof(x)=g(f(x))=g(3x1)=2(3x1)(3x1)2=6x2(9x26x+1)=6x29x2+6x1=9x2+12x3 gof(x)=9x2+12x3

Which is in the form of a natural polynomial

Therefore , the domain is (,) .

fof(x)=f(f(x))=f(3x1)=3(3x1)1=9x31=9x4 fof(x)=9x4

Domain is (,) .

gog(x)=g(g(x))=g(2xx2)=2(2xx2)(2xx2)2=4x2x2(4x24x3+x4)=4x2x24x2+4x3x4=x4+4x36x2+4x gog(x)=x4+4x36x2+4x

Domain is (,) .

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