+1206 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.
+128474 f(g(x)) = 3(ax + b) + 2 = 3ax + 3b + 2
g(f(x)) = a(3x + 2) + b = 3ax + 2a + b
Since these are equal then
3ax + 3b + 2 = 3ax + 2a + b
3b + 2 = 2a + b
2b + 2 = 2a
b + 1 = a
b = a - 1
And since ab = 20.....then
a (a - 1) = 20
a^2 - a = 20
a^2 - a - 20 = 0
(a - 5) (a + 4) = 0
Setting each factor to 0 and solving for a produces a = 5 or a = -4
So....the sum of these possible values for a = 1
