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In this work a geometric interpretation of the obstructions to the eXtension of functions obtained_ from intersection of functions, is given. Let M^m and Nⁿ be smooth closed manifolds land let V ⊂ M and K ⊂ N be closed submanifolds of same codimension. One of our goals is to give a necessary and sufficient condition for the existence of a smooth map f: M → N, transversal to K, such that V = f^-1(K). In Chapter I we obtain conditions for the non existence of the map f. In Chapter III we find some results that guarantee the existence of such a map. For example: if V^m-2 ⊃ M^m is an oriented submanifold homologous to zero in an oriented manifold M, then there exists f: M → Sⁿ such that f Φ s^n-2 and V^m-2 = f^-1 (S^n-2).