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 \(a\).
Post answer w/ explanation please.
\(a(3x+2)+b=3(ax+b)+2\\ 3ax+2a+b=3ax+3b+2\\ 2a+b=3b+2\\ 2a-2b=2\\ a-b=1\\ a-1=b\)
So as long as \(a-1=b\), \(f(g(x)) = g(f(x))\). The only other condition we need to satisfy is \(ab=20\), so:
\(a(a-1)=20\\ a^2-a-20=0\\ (a-5)(a+4)=0\\ a=5, a=-4\)
Sum them up and you'll get the answer