Purely kinetic k-essence models in which the Lagrangian contains only a kinetic factor and does not depend explicitly on the field itself are considered, and a theoretical constraint is obtained: F_x = F₀a⁻³. Under this theoretical constraint, we discuss a kind of purely k-essence with form F(X) = -(1 + 2Xⁿ)^1/2n, which can be considered as the generalized tachyon field, and find that this kind of k-essence is not likely a candidate of dark energy to describe the present accelerated expansion of the Universe. This is contrary to a previous suggestion that k-essence with such a form may be used to describe phantom cosmologies.