Khanh: Created page with "The '''inverse image''' (or '''preimage''') of a given subset $B$ of the codomain of $f,$ is the set of all elements of the domain that map to the member..."

2021-10-03T22:50:17Z

Created page with "The '''inverse image''' (or '''preimage''') of a given subset <math>B</math> of the codomain of <math>f,</math> is the set of all elements of the domain that map to the member..."

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The '''inverse image''' (or '''preimage''') of a given subset <math>B</math> of the codomain of <math>f,</math> is the set of all elements of the domain that map to the members of <math>B.</math>

==Inverse image==

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>. The inverse image of a singleton set, denoted by <math>f^{-1}[\{ y \}]</math> or by <math>f^{-1}[y],</math> is also called the 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>

Functions:Inverse Image - Revision history

Khanh: Created page with "The '''inverse image''' (or '''preimage''') of a given subset B of the codomain of f, is the set of all elements of the domain that map to the member..."

Khanh: Created page with "The '''inverse image''' (or '''preimage''') of a given subset $B$ of the codomain of $f,$ is the set of all elements of the domain that map to the member..."