2004 AMC 12B Problem 13

Since $f(f^{-1}(x)) = x,$ we have $a(bx + a) + b = x.$ Matching terms gives $ab = 1$ and $a^2 + b = 0.$ Then $b = \frac{1}{a}$ and $a^2 + \frac{1}{a} = 0,$ so $a^3 = -1,$ giving $a = -1$ and $b = -1.$ Thus $a + b = -2.$ Thus, the correct answer is A.