The preimage of 4,9 under the function f: R → R :f(x)=x² is the set -3,-2,+2,+3.

1967 [Academic Press], Francois Treves, Topological Vector Spaces, Distributions and Kernels, 2006, Dover, page 22,
The preimage of a neighborhood U of 0 in E must be a neighborhood of (0,x), since (0,x) is mapped into 0.

Previously, maps and hypermaps were constructed as the preimages of a segment joining two of the three critical values.

Loosely speaking, saying that a function f is one-way implies that given y (in the range of f) it is infeasible to find a preimage of y under f.

Preimage

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