Example Graph

Consider nodes $1$ through $6$ where $f(1) = 2$, $f(2) = 3$, $f(3) = 4$, $f(4) = 3$, $f(5) = 6$, $f(6) = 6$. Each node has exactly one outgoing edge. From node $1$: you go $1 \to 2 \to 3 \to 4 \to 3 \to 4 \to \ldots$ The tail is $1 \to 2$, and the cycle is $3 \to 4 \to 3$.

The tail has length $2$, the cycle has length $2$. From node $6$: you stay at $6$ forever. A self-loop is a cycle of length $1$. Every node eventually reaches a cycle, possibly immediately.