Inverse function: Difference between revisions

Revision as of 20:29, 3 November 2024

Revision as of 19:35, 19 December 2024

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If {{math|''f'': ''X'' → ''Y''}} is any function (not necessarily invertible), the '''preimage''' (or '''inverse image''') of an element {{math| ''y'' ∈ ''Y''}} is defined to be the set of all elements of {{mvar|X}} that map to {{mvar|y}}:

: <math>f^{-1}(\{y\}) = \left\{ x\in X : f(x) = y \right\} . </math>

The preimage of {{mvar|y}} can be thought of as the [[image (mathematics)|image]] of {{mvar|y}} under the (multivalued) full inverse of the function {{mvar|f}}.

SimilarlyThe notion can be generalized to subsets of the range. Specifically, if {{mvar|S}} is any [[subset]] of {{mvar|Y}}, the preimage of {{mvar|S}}, denoted by <math>f^{-1}(S) </math>, is the set of all elements of {{mvar|X}} that map to {{mvar|S}}:

: <math>f^{-1}(S) = \left\{ x\in X : f(x) \in S \right\} . </math>

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: <math>f^{-1}(\left\{1,4,9,16\right\}) = \left\{-4,-3,-2,-1,1,2,3,4\right\}</math>.

The original notion and its generalization are related by the identity <math>f^{-1}(y) = f^{-1}(\{y\}),</math> The preimage of a single element {{math| ''y'' &isin; ''Y''}} – a [[singleton set]] {{math|{''y''} }} – is sometimes called the ''[[fiber (mathematics)|fiber]]'' of {{mvar|y}}. When {{mvar|Y}} is the set of real numbers, it is common to refer to {{math|''f''⁻¹({''y''})}} as a ''[[level set]]''.

==See also==