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The purpose of this work is to study the finítely determined map-germs u : (R²,0) → (R,O) with respect to group Α of local coordinate transformations in both spaces. It consists of three essential parts: In the first part we analise the m-determination of some special germs. In the Second part we obtain the classification in Α-orbits and we show that there are infinite orbits for the 4-determined case . In the last part we study the germs u : (R²,0) → (R,0) with det Hu(0,0) = 0 and j²u(0,0) ≠ 0, and we obtain normal forms in several cases. In this dissertation we use mainly the technique of coordinate transformations and some Mather's results [6].