The Collatz conjecture is an open problem involving the function. The function belongs to a class of generalized
functions of relatively prime type. This paper focuses on exploring periodic cycles for an extension of a generalized
function of relatively prime type. By extending its domain to
, the result shows that every integer periodic point is isolated in the usual topology on
. Moreover, every positive integer periodic cycle for the extension is attracting if the generalized
function is satisfied by parameters under some conditions.