Editing Function (mathematics) (section)

=== Inverse and implicit functions ===
A function <math>f : X\to Y,</math> with domain {{mvar|X}} and codomain {{mvar|Y}}, is [[bijective]], if for every {{mvar|y}} in {{mvar|Y}}, there is one and only one element {{mvar|x}} in {{mvar|X}} such that {{math|1=''y'' = ''f''(''x'')}}. In this case, the [[inverse function]] of {{mvar|f}} is the function <math>f^{-1} : Y \to X</math> that maps <math>y\in Y</math> to the element <math>x\in X</math> such that {{math|1=''y'' = ''f''(''x'')}}. For example, the [[natural logarithm]] is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the [[exponential function]], that maps the real numbers onto the positive numbers.

If a function <math>f: X\to Y</math> is not bijective, it may occur that one can select subsets <math>E\subseteq X</math> and <math>F\subseteq Y</math> such that the [[restriction of a function|restriction]] of {{mvar|f}} to {{mvar|E}} is a bijection from {{mvar|E}} to {{mvar|F}}, and has thus an inverse. The [[inverse trigonometric functions]] are defined this way. For example, the [[cosine function]] induces, by restriction, a bijection from the [[interval (mathematics)|interval]] {{closed-closed|0, ''π''}} onto the interval {{closed-closed|−1, 1}}, and its inverse function, called [[arccosine]], maps {{closed-closed|−1, 1}} onto {{closed-closed|0, ''π''}}. The other inverse trigonometric functions are defined similarly.

More generally, given a [[binary relation]] {{mvar|R}} between two sets {{mvar|X}} and {{mvar|Y}}, let {{mvar|E}} be a subset of {{mvar|X}} such that, for every <math>x\in E,</math> there is some <math>y\in Y</math> such that {{math|''x R y''}}. If one has a criterion allowing selecting such a {{mvar|y}} for every <math>x\in E,</math> this defines a function <math>f: E\to Y,</math> called an [[implicit function]], because it is implicitly defined by the relation {{mvar|R}}.

For example, the equation of the [[unit circle]] <math>x^2+y^2=1</math> defines a relation on real numbers. If {{math|−1 < ''x'' < 1}} there are two possible values of {{mvar|y}}, one positive and one negative. For {{math|1=''x'' = ± 1}}, these two values become both equal to 0. Otherwise, there is no possible value of {{mvar|y}}. This means that the equation defines two implicit functions with domain {{closed-closed|−1, 1}} and respective codomains {{closed-open|0, +∞}} and {{open-closed|−∞, 0}}.

In this example, the equation can be solved in {{mvar|y}}, giving <math>y=\pm \sqrt{1-x^2},</math> but, in more complicated examples, this is impossible. For example, the relation <math>y^5+y+x=0</math> defines {{mvar|y}} as an implicit function of {{mvar|x}}, called the [[Bring radical]], which has <math>\mathbb R</math> as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and [[nth root|{{mvar|n}}th roots]].

The [[implicit function theorem]] provides mild [[differentiability]] conditions for existence and uniqueness of an implicit function in the neighborhood of a point.