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