section_path: Specifying a function/Inverse and implicit functions
section_name: Inverse and implicit functions
paragraph_text: Inverse and implicit functions
  
  A function f: X → Y, {\displaystyle f:X\to Y,} with domain X and codomain Y, is bijective, if for every y in Y, there is one and only one element x in X such that y = f (x). In this case, the inverse function of f is the function f − 1: Y → X {\displaystyle f^{-1}:Y\to X} that maps y ∈ Y {\displaystyle y\in Y} to the element x ∈ X {\displaystyle x\in X} such that 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 f: X → Y {\displaystyle f:X\to Y} is not bijective, it may occur that one can select subsets E ⊆ X {\displaystyle E\subseteq X} and F ⊆ Y {\displaystyle F\subseteq Y} such that the restriction of f to E is a bijection from E to 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 onto the interval, and its inverse function, called arccosine, maps onto. The other inverse trigonometric functions are defined similarly.
  
  More generally, given a binary relation R between two sets X and Y, let E be a subset of X such that, for every x ∈ E, {\displaystyle x\in E,} there is some y ∈ Y {\displaystyle y\in Y} such that x R y. If one has a criterion allowing selecting such a y for every x ∈ E, {\displaystyle x\in E,} this defines a function f: E → Y, {\displaystyle f:E\to Y,} called an implicit function, because it is implicitly defined by the relation R.
  
  For example, the equation of the unit circle x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} defines a relation on real numbers. If −1 < x < 1 there are two possible values of y, one positive and one negative. For x = ± 1, these two values become both equal to 0. Otherwise, there is no possible value of y. This means that the equation defines two implicit functions with domain and respective codomains.
  
  In this example, the equation can be solved in y, giving y = ± 1 − x 2, {\displaystyle y=\pm {\sqrt {1-x^{2}}},} but, in more complicated examples, this is impossible. For example, the relation y 5 + y + x = 0 {\displaystyle y^{5}+y+x=0} defines y as an implicit function of x, called the Bring radical, which has R {\displaystyle \mathbb {R} } as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and 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.