Editing Inverse function (section)

===Squaring and square root functions===
The function {{math|''f'': '''R''' → [0,∞)}} given by {{math|1=''f''(''x'') = ''x''<sup>2</sup>}} is not injective because <math>(-x)^2=x^2</math> for all <math>x\in\R</math>. Therefore, {{Mvar|f}} is not invertible.

If the domain of the function is restricted to the nonnegative reals, that is, we take the function <math>f\colon [0,\infty)\to [0,\infty);\ x\mapsto x^2</math> with the same ''rule'' as before, then the function is bijective and so, invertible.<ref>{{harvnb|Lay|2006|loc=p. 69, Example 7.24}}</ref> The inverse function here is called the ''(positive) square root function'' and is denoted by <math>x\mapsto\sqrt x</math>.
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===Inverses in higher mathematics===
The definition given above is commonly adopted in [[set theory]] and [[calculus]]. In higher mathematics, the notation
:<math>f\colon X \to Y </math>
means "{{mvar|f}} is a function mapping elements of a set {{mvar|X}} to elements of a set {{mvar|Y }}". The source, {{mvar|X}}, is called the domain of {{mvar|f}}, and the target, {{mvar|Y}}, is called the [[codomain]]. The codomain contains the range of {{mvar|f}} as a [[subset]], and is part of the definition of {{mvar|f}}.<ref>{{harvnb|Smith|Eggen|St. Andre|2006|loc=p. 179}}</ref>

When using codomains, the inverse of a function {{math| ''f'': ''X'' → ''Y''}} is required to have domain {{mvar|Y}} and codomain {{mvar|X}}. For the inverse to be defined on all of {{mvar|Y}}, every element of {{mvar|Y}} must lie in the range of the function {{mvar|f}}. A function with this property is called ''onto'' or ''[[Surjective function|surjective]]''. Thus, a function with a codomain is invertible if and only if it is both ''[[Injective function|injective]]'' (one-to-one) and surjective (onto). Such a function is called a one-to-one correspondence or a [[bijection]], and has the property that every element {{math| ''y'' ∈ ''Y''}} corresponds to exactly one element {{math| ''x'' ∈ ''X''}}.
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