AMC 12 Mock Exam 1 Problem 6

We have $f(f(x))=a(ax+b)+b=a^2x+(a+1)b$. Matching coefficients with $4x+9$ gives $a^2=4$ and $(a+1)b=9$.

If $a=2$, then $3b=9$, so $b=3$ and $f(1)=5$. If $a=-2$, then $-b=9$, so $b=-9$ and $f(1)=-2-9=-11$. These are the only possibilities, so the sum of all possible values of $f(1)$ is $5+(-11)=\boxed{-6}$.