To protect data in cloud servers, fully homomorphic encryption (FHE) is an effective solution. In addition to encrypting data, FHE allows a third party to evaluate arithmetic circuits (i.e., computations) over encrypted data without decrypting it, guaranteeing protection even during the calculation. However, FHE supports only addition and multiplication. Functions that cannot be directly represented by additions or multiplications cannot be evaluated with FHE. A naïve implementation of such arithmetic operations with FHE is a bit-wise operation that encrypts numerical data as a binary string. This incurs huge computation time and storage costs, however. To overcome this limitation, we propose an efficient protocol to evaluate multi-input functions with FHE using a lookup table. We extend our previous work, which evaluates a single-integer input function, such as f(x). Our extended protocol can handle multi-input functions, such as f(x,y). Thus, we propose a new method of constructing lookup tables that can evaluate multi-input functions to handle general functions. We adopt integer encoding rather than bit-wise encoding to speed up the evaluations. By adopting both permutation operations and a private information retrieval scheme, we guarantee that no information from the underlying plaintext is leaked between two parties: a cloud computation server and a decryptor. Our experimental results show that the runtime of our protocol for a two-input function is approximately 13 minutes, when there are 8,192 input elements in the lookup table. By adopting a multi-threading technique, the runtime can be further reduced to approximately three minutes with eight threads. Our work is more practical than a previously proposed bit-wise implementation, which requires 60 minutes to evaluate a single-input function.