stylistic:tone, stylistic:style_transfer
Partial inverses

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 x = (− x). 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 (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 + 2 π n) = 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 − π / 2 and π / 2. The following table describes the principal branch of each inverse trigonometric function: