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

Jul 24, 2017

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Find  c  if  f( g(x) )  =  g( f(x) )  for all  x  .

f(x)  =  2x + 7                          To find  f( g(x)  ) , replace every instance of  x  with  g(x)  .

f( g(x) )  =  2( g(x) ) + 7             Since  g(x)  =  3x + c  , we can write...

f( g(x) )  =  2( 3x + c ) + 7

g(x)  =  3x + c                         To find  g( f(x) ) , replace every instance of  x  with  f(x)  .

g( f(x) )  =  3( f(x) ) + c              Since  f(x)  =  2x + 7  , we can write...

g( f(x) )  =  3( 2x + 7 ) + c

We want to know what  c  is when

f( g(x) )  =  g( f(x) )                                   Substitute the functions in.

2( 3x + c ) + 7  =  3( 2x + 7 ) + c

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

2c + 7  =  21 + c                                    Subtract  c  from both sides, and subtract  7  from both sides.

c  =  14

Jul 24, 2017