BuyFind

Precalculus: Mathematics for Calcu...

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

Precalculus: Mathematics for Calcu...

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

Solutions

Chapter 2.6, Problem 43E
To determine

To find:The functions fog , gof , fof , gog and their domains for f(x)=xx+1 and g(x)=1x .

Expert Solution

Answer to Problem 43E

The function fog is equal to 11+x with domain set of real numbers except x=1 and x=0 , gof is equal to x+1x with domain set of real numbers except x=1 and x=0 , fof is equal to x2x+1 with domain set of real number except x=1 and x=12 , gog is equal to (gog)(x)=x with domain set of real numbers except x=0 for f(x)=xx+1 and g(x)=1x .

Explanation of Solution

Given information:

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

Calculation:

Simplify the function fog .

  (fog)(x)=f(g(x))=f(1x)=1x1x+1=11+x

The function g(x)=1x is defined for all the real values of x except x=0 and the function (fog)(x)=11+x is defined only if 1+x0 i.e. x1 .

So, the domain of the function (fog)(x)=11+x is set of real numbers except x=1 and x=0 .

Simplify the function gof .

  (gof)(x)=g(f(x))=g(xx+1)=1xx+1=x+1x

The function f(x)=xx+1 is defined for all the real values of x except x=1 as the denominator becomes zero and the function (gof)(x)=x+1x is defined only if x0 .

So, the domain of the function (gof)(x)=x+1x is set of real numbers except x=1 and x=0 .

Simplify the function fof .

  (fof)(x)=f(f(x))=f(xx+1)=xx+1xx+1+1

Further simplify,

  (fof)(x)=xx+1xx+1+1=xx+1x+x+1x+1=x2x+1

The function f(x)=xx+1 is defined for all the real values of x except x=1 and the function (fof)(x)=x2x+1 is defined only if 2x+10 .

So, the domain of the function (fof)(x)=x2x+1 is set of real number except x=1 and x=12 .

Simplify the function gog .

  (gog)(x)=g(g(x))=g(1x)=11x=x

The function g(x)=1x is defined for all the real values of x except x=0 and the function (gog)(x)=x is defined for all the real values of x . Combine both the conditions.

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

Therefore, the function fog is equal to 11+x with domain set of real numbers except x=1 and x=0 , gof is equal to x+1x with domain set of real numbers except x=1 and x=0 , fof is equal to x2x+1 with domain set of real number except x=1 and x=12 , gog is equal to (gog)(x)=x with domain set of real numbers except x=0 for f(x)=xx+1 and g(x)=1x .

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