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Let f(x)=x+1 and g(x)=2x. Also denote the inverses to these functions as f^-1 and g^-1. Compute \( \[f(g^{-1}(f^{-1}(f^{-1}(g(f(5)))))).\]\)

 Jul 13, 2019
 #1
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find f^-1 =                y = x+1    solve for x and then switch x and y's     x = y-1     so  f^-1 = y = x-1

 

find g^-1                  y = 2x    solve for x   x = y/2    switch y's and x's    y = x/2  = g^-1

 

 

Now f(5) (this is the original f(x) function....not thte inverse)   f(5) = 5+1 = 6

  then  g(6) = 2x = 2(6) = 12

   then f^-1  (12) = 11

    then  f^-1 (11) = 10

      then g^-1 (10 ) = 5

       then f (7) =

 Jul 14, 2019
 #3
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Problems posting   (internal server error)  ... corrected last step in answer below....   ~EP

ElectricPavlov  Jul 14, 2019
edited by ElectricPavlov  Jul 14, 2019
edited by ElectricPavlov  Jul 14, 2019
 #2
avatar+24571 
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find f^-1 =                y = x+1    solve for x and then switch x and y's     x = y-1     so  f^-1 = y = x-1

 

find g^-1                  y = 2x    solve for x   x = y/2    switch y's and x's    y = x/2  = g^-1

 

 

Now f(5) (this is the original f(x) function....not thte inverse)   f(5) = 5+1 = 6

  then  g(6) = 2x = 2(6) = 12

   then f^-1  (12) = 11

    then  f^-1 (11) = 10

      then g^-1 (10 ) = 5

       then f (5) = 6

 Jul 14, 2019
edited by ElectricPavlov  Jul 14, 2019
edited by ElectricPavlov  Jul 14, 2019

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