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

f(x) = ax + b

x = a*f^-1(x) + b

f^-1(x) = (1/a) * x - b/a

so you get 1/a = b, -b/a = a

because -b/a = a, b = -a^2

Only solution is a = -1, b = -1.

so a + b = -2.