Chase an inverse function backward

The prompt

Suppose g(x) = f⁻¹(x), g(0) = 5, g(4) = 7, g(3) = 2, g(7) = 9, g(9) = 6, what is f(f(f(6)))?

Expected result

A step-by-step inverse-function chase: since g=f⁻¹, f(x)=y whenever g(y)=x. Working inward: f(6)=9 (from g(9)=6), f(9)=7 (from g(7)=9), f(7)=4 (from g(4)=7) → f(f(f(6)))=4.

Why it works

Tests whether a model correctly uses the inverse-function relationship (g=f⁻¹) to work an unfamiliar nested-function problem backward, rather than needing f defined explicitly.