page_title: Inverse function
section_title: Partial inverses
hierarchical_section_title: Inverse function / Generalizations / Partial inverses
caption_attribution_description: Català: La inversa d'aquesta funció cúbica té tres branques.English: The inverse of this cubic function has three branches.
context_page_description: In mathematics, an inverse function is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., f(x) = y if and only if g = x.
  As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure, to reverse this and get x back from some output value, say y, we should undo each step in reverse order. In this case it means we should add 7 to y and then divide the result by 5. In functional notation this inverse function would be given by,
  With y = 5x − 7 we have that f(x) = y and g = x.
  Not all functions have inverse functions. Those that do are called invertible. For a function f: X → Y to have an inverse, it must have the property that for every y in Y there is one, and only one x in X so that f(x) = y. This property ensures that a function g: Y → X exists with the necessary relationship with f.
context_section_description: Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain. For example, the function
  is not one-to-one, since x² = (−x)². However, the function becomes one-to-one if we restrict to the domain  x ≥ 0, in which case
  (If we instead restrict to the domain  x ≤ 0, then the inverse is the negative of the square root of y.)  Alternatively, there is no need to restrict the domain if we are content with the inverse being a multivalued function:
  Sometimes this multivalued inverse is called the full inverse of f, and the portions (such as √x and −√x) are called branches. The most important branch of a multivalued function (e.g. the positive square root) is called the principal branch, and its value at y is called the principal value of f ⁻¹(y).
  For a continuous function on the real line, one branch is required between each pair of local extrema. For example, the inverse of a cubic function with a local maximum and a local minimum has three branches (see the adjacent picture).
  These considerations are particularly important for defining the inverses of trigonometric functions. For example, the sine function is not one-to-one, since
  for every real x (and more generally sin(x + 2πn) = sin(x) for every integer n). However, the sine is one-to-one on the interval
  [−, ], and the corresponding partial inverse is called the arcsine. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between − and . The following table describes the principal branch of each inverse trigonometric function:
caption: The inverse of this cubic function has three branches.
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