 | Function mathematics: Encyclopedia II - Function mathematics - Image of a set
Function mathematics - Image of a set
One often extends the concept (and notation) of image of an argument to sets of arguments. Namely, if A is any subset of the domain X, the image of A under f is the subset of Y defined
f(A) = {f(x) | x is in A}
So, for example, the image of {-3,2,3} under the squaring function sqr is sqr({-3,2, 3}) = {4, 9}.
This extension is consistent as long as no subset of the domain is also an element of the domain. A few authors write f[A] instead of f(A), to emphasize the distinction between the two concepts; a few others write f` x instead of f(x), and f``A instead of f[A].
Function mathematics - Range of a function
The set of all possible values of a function f; that is, the set of all y such that there exists a pair (x,y) in the graph G of f, is most commonly called the range of f. The range is a subset of the codomain, but may be different from it. For example, the range of the squaring function defined above is the set of all perfect squares, which is a proper subset of the codomain N, the natural numbers.
On the other hand, the symmetric concept, the set of all x such that there is a pair (x, y) in the graph of f, always coincides with the domain of f.
Notice that the range of f is the image f(X) of its domain.
Function mathematics - Preimage of a set
The preimage (or inverse image) of a subset B of the codomain Y under a function f is the subset of the domain X defined by
f −1(B) = {x in X | f(x) is in B}
So, for example, the preimage of {4,5,9} under sqr is the set {-3,-2,+2,+3}.
In general, the preimage of a singleton set (a set with exactly one element) may have zero elements, or more than one. For example, the preimage of {4} under sqr is the set {-2,+2}, and the preimage of {5} is the empty set ∅. For this reason, one cannot in general define the preimage of an element of Y as being an element of X. However, one usually makes the convention that f −1(b) = f −1({b}), i. e.
f −1(b) = {x in X | f(x) = b}
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 Adapted from the Wikipedia article "Image of a set", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
