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In this case, <math>g</math> is called a [[Retract (category theory)|retraction]] of <math>f.</math> Conversely, <math>f</math> is called a [[Retract (category theory)|section]] of <math>g.</math>
For example: <math>f:\R\rightarrow\R^2,x\mapsto(1,m)^\intercal x</math> is retracted by <math>g:y\mapsto\frac{(1,m)}{1+m^2}y</math>.
Conversely, every injection <math>f</math> with a non-empty ___domain has a left inverse <math>g</math>. It can be defined by choosing an element <math>a</math> in the ___domain of <math>f</math> and setting <math>g(y)</math> to the unique element of the pre-image <math>f^{-1}[y]</math> (if it is non-empty) or to <math>a</math> (otherwise).{{refn|Unlike the corresponding statement that every surjective function has a right inverse, this does not require the [[axiom of choice]], as the existence of <math>a</math> is implied by the non-emptiness of the ___domain. However, this statement may fail in less conventional mathematics such as [[constructive mathematics]]. In constructive mathematics, the inclusion <math>\{ 0, 1 \} \to \R</math> of the two-element set in the reals cannot have a left inverse, as it would violate [[Indecomposability (constructive mathematics)|indecomposability]], by giving a [[Retract (category theory)|retraction]] of the real line to the set {0,1}.}}
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