What is the purpose of a trapdoor one$-$way function in public key cryptography?

Question $4$ options:

To encrypt data transmissions over a network to prevent eavesdropping.

To ensure that the encryption process is reversible only by using a secret key, known only to the recipient.

To provide a method for generating random public keys that cannot be predicted.

To validate the integrity and origin of a message through digital signatures without encryption.