Permutations are always invertible. Otherwise several different inputs will be mapped to the same output, which needlessly reduces your state space.

The inverse is also required for decryption: If you encrypt as ciphertext = permutation(plaintext + key), decryption is plaintext = inverse(ciphertext) - key.

My question is why does Keccak not use a one-way function instead of a permutation.

> The inverse is also required for decryption: If you encrypt as ciphertext = permutation(plaintext + key), decryption is plaintext = inverse(ciphertext) - key.

This is not how encryption/decryption works with Keccak. Keccak is used to create a stream (it is then XORed with the plaintext or ciphertext).