Let \(f(x)=3x+2\) and \(g(x)=ax+b\), for some constants \(a \) and \(b\). If \(ab=20\) and \(f(g(x))=g(f(x))\) for \(x=0,1,2\ldots 9\), find the sum of all possible values of \(a\).
For f(g(x)) you'll get 3ax+3b+2 and for g(f(x)) you'll get 3ax+2a+b. The equation 3ax+3b+2=3ax+2a+b can be simplified to b+1=a. We also know that ab=20 so the only value where ab=20 and b+1=a is where b=4 and a=5. So your answer is 5.