section_path: General properties/Image and preimage
section_name: Image and preimage
paragraph_text: Image and preimage
  
  Let f: X → Y. {\displaystyle f:X\to Y.} The image under f of an element x of the domain X is f (x). If A is any subset of X, then the image of A under f, denoted f (A), is the subset of the codomain Y consisting of all images of elements of A, that is,
  
  The image of f is the image of the whole domain, that is, f (X). It is also called the range of f, although the term range may also refer to the codomain.
  
  On the other hand, the inverse image or preimage under f of an element y of the codomain Y is the set of all elements of the domain X whose images under f equal y. In symbols, the preimage of y is denoted by f − 1 (y) {\displaystyle f^{-1}(y)} and is given by the equation
  
  Likewise, the preimage of a subset B of the codomain Y is the set of the preimages of the elements of B, that is, it is the subset of the domain X consisting of all elements of X whose images belong to B. It is denoted by f − 1 (B) {\displaystyle f^{-1}(B)} and is given by the equation
  
  For example, the preimage of { 4, 9 } {\displaystyle \{4,9\}} under the square function is the set { − 3, − 2, 2, 3 } {\displaystyle \{-3,-2,2,3\}}.
  
  By definition of a function, the image of an element x of the domain is always a single element of the codomain. However, the preimage f − 1 (y) {\displaystyle f^{-1}(y)} of an element y of the codomain may be empty or contain any number of elements. For example, if f is the function from the integers to themselves that maps every integer to 0, then f − 1 (0) = Z {\displaystyle f^{-1}(0)=\mathbb {Z} }.
  
  If f: X → Y {\displaystyle f:X\to Y} is a function, A and B are subsets of X, and C and D are subsets of Y, then one has the following properties:
  
  The preimage by f of an element y of the codomain is sometimes called, in some contexts, the fiber of y under f.
  
  If a function f has an inverse (see below), this inverse is denoted f − 1. {\displaystyle f^{-1}.} In this case f − 1 (C) {\displaystyle f^{-1}(C)} may denote either the image by f − 1 {\displaystyle f^{-1}} or the preimage by f of C. This is not a problem, as these sets are equal. The notation f (A) {\displaystyle f(A)} and f − 1 (C) {\displaystyle f^{-1}(C)} may be ambiguous in the case of sets that contain some subsets as elements, such as { x, { x } }. {\displaystyle \{x,\{x\}\}.} In this case, some care may be needed, for example, by using square brackets f, f − 1 {\displaystyle f,f^{-1}} for images and preimages of subsets and ordinary parentheses for images and preimages of elements.