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

Let me just rewrite the details in a more understandable manner

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

Proceeding to the problem, we have to find the value of c by making use of the relationship written by the third equation. The notation f( g(x) ) means that you have to replace x with g(x). So, you would have incorporate one equation to the other, and vice versa. By equating them, we can determine x. Let us start with f( g(x) ) .

f( g(x) ) = 2(3x + c) + 7
f( g(x) ) = 6x + 2c + 7

Next, we do the similar thing to g( f(x) ).
g( f(x) ) = 3(2x+7) + c
g( f(x) ) = 6x + 21 + c

Then, we equate both simplified equations
6x + 2c + 7 = 6x + 21 + c

Let's simplify further by placing c on one side, and the rest on the other side:
2c - c = 6x - 6x + 21 - 7
c = 14