For generalized 3

x

+1 function

T

(

x

)

the feature of fixed points and their existence-domain analysis is an important problem in fractal.

T

(

x

) is a complex transcendental function and its fixed point in C-plane is hardly to solve.Meanwhile

the feature of fixed point is difficult to analyze.All these become an obstacle for the further study of

T

(

x

) dynamic system.In this paper

Because of the topological invariance of

T

(

x

)

we constructively proved its fixed point in C-plane firstly.Then we give the analysis for the existence domain of fixed points as well as their feature.Based on the existence do

main of

T

(

x

) fixed points

we estimated their distributions in C-plane.So we put forward a numerical algorithm for solving the fixed points of

T

(

x

) by analyzed the basis of distributions.Furthermore

we obtained some convergence domains of

T

(

x

) in C-plane and drew fractal image of these domains.The result of numerical experiment shows that the algorithm in this paper is correct and easy to implement.