# What is the range of a universal turing machine (partial function)?

I take the definition from Computability and Complexity Theory by Steven Homer and Alan L. Selman, 2011.

In their book they said that the domain of a computably enumerable (c.e.) function (f) can be the range of another partial function (p). In particular, the partial function can be the universal Turing machine. The universal Turing machine takes a machine description and input word (e, x) and effectively maps the domain (S) of the c.e. function (f).

My question is : what is the range of the universal Turing machine - shouldn't it just accept or reject ? then how could it map the domain of the c.e. function (f)....

Note that recognising words from a language can be thought of as computing a function from $\Sigma^* \rightarrow \{0,1\}$, so you don't have to accept via declaring some state accepting.