Image (mathematics)

In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. The image of a subset A ⊆ X under f is the subset of Y defined by
:f(A) = {y ∈ Y | y = f(x) for some x ∈ A}
Notice that the range of f is the image f(X) of its domain X.

With this definition, the image f becomes a function whose domain is the set of all subsets of X (also known as the power set of X) and whose codomain is the power set of Y. Note that the same notation is used for the original function f and its image. This is a common convention; the intended usage must be inferred by context.

The preimage or inverse image of a set B ⊆ Y under f is the subset of X defined by
:f⁻¹(B) = {x ∈ X | f(x) ∈ B}

Note that f⁻¹ should not be confused with the inverse function. The two only coincide if f is bijective. f⁻¹ is a new function whose domain is the power set of Y and whose codomain is the power set of X.

Example

1. f: {1,2,3} → {a,b,c,d} defined by

In this example, the image of {2,3} under f is f({2, 3}) = {d, c} and the range of f is {a, d, c}. The preimage of {a, c} is f⁻¹({a, c}) = {1,3}.

2. f: R → R defined by f(x)=x².

In this example, the image of {-2,3} under f is f({-2,3})={4,9} and the range of f is the set of nonnegative real numbers. The preimage of {4,9} under f is f⁻¹({4,9})={-2,2,-3,3}.

Consequences

Some consequences that follow immediately from these definitions are:

f(A₁ ∪ A₂) = f(A₁) ∪ f(A₂)

f(A₁ ∩ A₂) ⊆ f(A₁) ∩ f(A₂)

f⁻¹(B₁ ∪ B₂) = f⁻¹(B₁) ∪ f⁻¹(B₂)

f⁻¹(B₁ ∩ B₂) = f⁻¹(B₁) ∩ f⁻¹(B₂)

f(f⁻¹(B)) ⊆ B

f⁻¹(f(A)) ⊇ A

A₁ ⊆ A₂ implies f(A₁) ⊆ f(A₂)

B₁ ⊆ B₂ implies f⁻¹(B₁) ⊆ f⁻¹(B₂)

These are valid for arbitrary subsets A, A₁ and A₂ of the domain and arbitrary subsets B, B₁ and B₂ of the codomain.
The results relating images and preimages to the algebra of intersection and union work for any collections of subsets, not just for pairs of subsets.