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Combination and Composition numbers

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!
 

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Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
4.8 4.8 (4 lesson ratings) (4)
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 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
 
Steve S. | Tutoring in Precalculus, Trig, and Differential CalculusTutoring in Precalculus, Trig, and Diffe...
5.0 5.0 (3 lesson ratings) (3)
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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.
Crystal H. | Chemistry and English tutorChemistry and English tutor
4.7 4.7 (91 lesson ratings) (91)
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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