By analyzing the examples given, solvers can find that f(a, b, c) is a function that removes b from and adds c to a and g(a, b) is a function that counts the number of b in a.
Based on the last examples of f and g, solvers should determine that Z is a head. Specifically, a rock with a head is 🗿, and a coin has one head.
Then, the last set of equations indicates that X has nine heads, and removing a head from X results in X again. They are also given that the number of heads of this object after a head has been removed is 10.
The fact that X has 9 heads and has 10 heads after one is removed along with the title of the puzzle should clue that X is the Hydra, a monster that Hercules fought.
The last two equations in this block say that removing a head of the Hydra with Y results in the Hydra again, but Hydra now has 8 heads. According to Greek lore, Hercules used fire to remove a head from the Hydra permanently. Thus, Y is fire.
Finally, we can construct the answer: Y is FIRE, X is HYDRA, and f(🐜, 🅰️, ∅) is removing A from ANT, so our answer is FIRE HYDRANT.