Examples
1. f: {1,2,3} → {a,b,c,d} defined by
The image of the set {2,3} under f is f({2,3}) = {a,c}. The image of the function f is {a,c}. The preimage of a is f −1({a}) = {1,2}. The preimage of {a,b} is also {1,2}. The preimage of {b,d} is the empty set {}.
2. f: R → R defined by f(x) = x2.
The image of {-2,3} under f is f({-2,3}) = {4,9}, and the image of f is R+. The preimage of {4,9} under f is f −1({4,9}) = {-3,-2,2,3}. The preimage of set N = {n ∈ R | n < 0} under f is the empty set, because the negative numbers do not have square roots in the set of reals.
3. f: R2 → R defined by f(x, y) = x2 + y2.
The fibres f −1({a}) are concentric circles about the origin, the origin itself, and the empty set, depending on whether a>0, a=0, or a<0, respectively.
4. If M is a manifold and π :TM→M is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces Tx(M) for x∈M. This is also an example of a fiber bundle.
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