Let f(x)=3x+2 and g(x)=ax+b, for some constants a and b. If ab=90 and f(g(x))=g(f(x)) for x=0,1,2,3,4,5,6,7,8,9, find all possible values of a. Thanks
+23252 f(x) = 3x + 2
g(x) = ax + b
f( g(x) ) = 3( ax + b ) + 2 = 3ax + 3b + 2
g( f(x) ) = a( 3x + 2 ) + b = 3ax + 2a + b
Since f( g(x) ) = g( f(x) ) ---> 3ax + 3b + 2 = 3ax + 2a + b
3b + 2 = 2a + b
2b + 2 = 2a
b + 1 = a
Since ab = 90 ---> a = 90 / b
Substituting: b + 1 = a ---> b + 1 = 90/b
b2 + b = 90
b2 + b - 90 = 0
(b + 10)(b - 9) = 0
If b = -10, a = -9.
If b = 9, a = 10
