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

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

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