How do you use RSA for both authentication and secrecy?

Ofcourse Yes!We can use RSA for both authentication&secrecy purpose.

RSA is based upon public key/private key concept. For authentication one can encrypt the hash (MD5/SHA) of the data with his private key. This is known as digital signature. And secrecy is achieved by encrypting the data with the public key of the target user. Generally we dont use RSA for encryption because of key size (1024 bits). Rather a symmetric session key (128/256 bit) is established between communicating parties and is used for encryption.

RSA uses PKI that is correct - for authentication RSA is used int he form of X.509 Certificate. Your answer of using Hash/Digital Signature is the answer if the question was "non-repudiation"