Composition of Inverses

If the inverse function is to be defined over all real numbers then $f(x)$ itself must be continuous. Thus we require that $x + a = 3x - 1$ at $x = 8$ , which is the case when $a = 15.$

Thus $f^{-1}(x) = x - 15$ for $x \le 23$ and $f^{-1}(x) = \dfrac{1}{3}(x + 1)$ for $x \ge 23.$

We then have that $f^{-1}(65) = \dfrac{1}{3}(65 + 1) = 22$ , and

$f^{-1}(f^{-1}(65)) = f^{-1}(22) = 22 - 15 = \boxed{7}.$

Interesting thing of this question is to realise that if inverse function is defined for every real value, then function itself must be monotonous and continues. So a=15 . Rest thing is calculation. Final task is to find out the value of x such that f(x)=22 . Which gives x=7 . Ans.