Editing Image (mathematics) (section)

==Inverse image==

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Let <math>f</math> be a function from <math>X</math> to <math>Y.</math> The '''preimage''' or '''inverse image''' of a set <math>B \subseteq Y</math> under <math>f,</math> denoted by <math>f^{-1}[B],</math> is the subset of <math>X</math> defined by
<math display="block">f^{-1}[ B ] = \{ x \in X \,:\, f(x) \in B \}.</math>

Other notations include <math>f^{-1}(B)</math> and <math>f^{-}(B).</math>{{sfn|Dolecki|Mynard|2016|pp=4-5}} 
The inverse image of a [[Singleton (mathematics)|singleton set]], denoted by <math>f^{-1}[\{ y \}]</math> or by <math>f^{-1}(y),</math> is also called the [[Fiber (mathematics)|fiber]] or fiber over <math>y</math> or the [[level set]] of <math>y.</math> The set of all the fibers over the elements of <math>Y</math> is a family of sets indexed by <math>Y.</math>

For example, for the function <math>f(x) = x^2,</math> the inverse image of <math>\{ 4 \}</math> would be <math>\{ -2, 2 \}.</math> Again, if there is no risk of confusion, <math>f^{-1}[B]</math> can be denoted by <math>f^{-1}(B),</math> and <math>f^{-1}</math> can also be thought of as a function from the power set of <math>Y</math> to the power set of <math>X.</math> The notation <math>f^{-1}</math> should not be confused with that for [[inverse function]], although it coincides with the usual one for bijections in that the inverse image of <math>B</math> under <math>f</math> is the image of <math>B</math> under <math>f^{-1}.</math>