Step-by-Step Solution
Key Concept: $f(x) = f^{-1}(x)$ iff $f(f(x)) = x$ or $f(x) = x$. On the restricted domain $[1,5]$, the equation $f(x) = x$ gives the fixed points. Count fixed points by analysing each piecewise branch in $[1,5]$.
The fixed points in $[1,5]$ are $x = 1, 2, 3, 4, 5$ (5 values). Additionally, 4 values of $x$ satisfy $f(x) = y \neq x$ with $f(y) = x$, giving a total of $5 + 4 = 9$ values.
Correct Answer: 9