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 Let f(x) = 2x + 7 and g(x) = 3x + c. Find c if (f @ g)(x) = (g @ f)(x) for all x.

 

Thanks :D

 Nov 6, 2017

Best Answer 

 #1
avatar+7350 
+1

f(x)  =  2x + 7

g(x)  =  3x + c

 

f( g(x) )  =  f( 3x + c )  =  2( 3x + c ) + 7

 

g( f(x) )  =  g( 2x + 7 )  =  3( 2x + 7 ) + c

 

And we want to know what  c  is when  f( g(x)  )  =  g( f(x) )  . That is... when

 

2( 3x + c ) + 7  =  3( 2x + 7 ) + c           Distribute the numbers outside the parenthesees.

 

6x + 2c + 7  =  6x + 21 + c                   Subtract  6x  from both sides of the equation.

 

2c + 7  =  21 + c             Subtract  c  from both sides of the equation.

 

c  + 7  =  21                    Subtract  7  from both sides of the equation.

 

c  =  14

 Nov 6, 2017
 #1
avatar+7350 
+1
Best Answer

f(x)  =  2x + 7

g(x)  =  3x + c

 

f( g(x) )  =  f( 3x + c )  =  2( 3x + c ) + 7

 

g( f(x) )  =  g( 2x + 7 )  =  3( 2x + 7 ) + c

 

And we want to know what  c  is when  f( g(x)  )  =  g( f(x) )  . That is... when

 

2( 3x + c ) + 7  =  3( 2x + 7 ) + c           Distribute the numbers outside the parenthesees.

 

6x + 2c + 7  =  6x + 21 + c                   Subtract  6x  from both sides of the equation.

 

2c + 7  =  21 + c             Subtract  c  from both sides of the equation.

 

c  + 7  =  21                    Subtract  7  from both sides of the equation.

 

c  =  14

hectictar Nov 6, 2017
 #2
avatar+98125 
+1

This is an interesting problem......thanks, hectictar......!!!!

 

cool cool cool

 Nov 6, 2017

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