AMC 12 2004b Problem 13

If $f(x) = ax+b$ and $f^{-1}(x) = bx+a$ with $a$ and $b$ real, what is the value of $a+b$?

$\mathrm{(A)}\ -2 \mathrm{(B)}\ -1 \mathrm{(C)}\ 0 \mathrm{(D)}\ 1 \mathrm{(E)}\ 2$