Combination and Composition numbers

Question

f(x) =X-2     g(X)=x^2+1

f(x) = X-2     (f+g) 2       ( f g )-4            (fog )2            (fof) 3           (fogof)X

Somebody please save me!

 

Answer 1

Given: f(x) = x - 2,  g(x) = x^2 + 1
Find: (f + g)(2),  (f • g)(-4),  (f o g)(2),  (f o f)(3),  (f o g o f)(x)

 

(f + g)(2) = f(2) + g(2) = (2 - 2) + (2^2 + 1) = 0 + 5 = 5

 

(f • g)(-4) = ( f(-4) ) ( g(-4) ) = (-4 - 2) ( (-4)^2 + 1 ) = -6(17) = -102

 

(f o g)(2) = f( g(2) ) = f( 2^2 + 1 ) = f(5) = 5 - 2 = 3

 

(f o f)(3) = f( f(3) ) = f( 3 - 2 ) = f(1) = 1 - 2 = -1

 

(f o g o f)(x) = f( g( f(x) ) ) = f( g( x - 2 ) ) = f( (x - 2)^2 + 1 )

= ( (x - 2)^2 + 1 ) - 2 = (x - 2)^2 - 1 = x^2 - 4x + 4 - 1 = x^2 - 4x + 3

 

This is an exercise in Functional Notation; learn it well. You will use all these operations in calculus. Composition of functions is used in the very important and key Chain Rule of Derivatives.

Answer 2

 f( X) = X - 2     g(x) = X^2 +1

 

 

(f + g)(X) = X^2 +1 +X -2 = X^2 + X -1

 

  ( f + g ) ( 2) = 2^2 + 2 - 1 = 5

 

  (f . g)(X)  = ( X- 2) ( X^2 + 1)

                 =  X^3 - 2 X^2 +X - 2

 

 ( f . g )(- 4 )  = (-4 ^ 3  ) - 2 ( -4^2) + (-4) - 2

                     = - 64  - 32 -4 -2

                      = - 102

 

  ( f . f ) = ( X -2 ) ^2 = X^2 - 4X +4

 

   ( f.f ) ( 3 ) = f( f(3)) =f(1) = 1 -2 = -1

 

     ( f . g . f ) =f(g(f(x)) =  f ( g( x-2) = f ( ( X-2)^2 +1) = ( X^2 -4x +5 -2) = X^2 -4X +3

 

Answer 3

Hi Carli.  You can think of these as substitution problems.

 

f(x) = x-2

g(x)=x^2+1

 

So for (fog)-4, you will start with your f(x) equation, f(x)=x-2.  However, wherever you see an x in your f(x) equation, you will substitute it with whatever your g(x) equation is equal to:

 

so since f(x)= (x)-2

(fog) = (x^2+1) -2

and to solve (fog)-4, you will plug -4 into your (fog) equation:

 

(fog)-4 = ((-4^2)+1))-2 = 15