Partial inverses Even if a function is not one-to-one, it may be possible to define a
partial inverse of by
restricting the domain. For example, the function : f(x) = x^2 is not one-to-one, since . However, the function becomes one-to-one if we restrict to the domain , in which case : f^{-1}(y) = \sqrt{y} . (If we instead restrict to the domain , then the inverse is the negative of the square root of .)
Full inverses has three branches. Alternatively, there is no need to restrict the domain if we are content with the inverse being a
multivalued function: : f^{-1}(y) = \pm\sqrt{y} . Sometimes, this multivalued inverse is called the
full inverse of , and the portions (such as and −) 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 is called the
principal value of . 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).
Trigonometric inverses is a partial inverse of the
sine function. The above considerations are particularly important for defining the inverses of
trigonometric functions. For example, the
sine function is not one-to-one, since : \sin(x + 2\pi) = \sin(x) for every real (and more generally for every
integer ). 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:
Left and right inverses Function composition on the left and on the right need not coincide. In general, the conditions • "There exists such that " and • "There exists such that " imply different properties of . For example, let denote the squaring map, such that for all in , and let denote the square root map, such that for all . Then for all in ; that is, is a right inverse to . However, is not a left inverse to , since, e.g., .
Left inverses If , a
left inverse for (or
retraction of ) is a function such that composing with from the left gives the identity function g \circ f = \operatorname{id}_X\text{.} That is, the function satisfies the rule : If , then . The function must equal the inverse of on the image of , but may take any values for elements of not in the image. A function with nonempty domain is injective if and only if it has a left inverse. An elementary proof runs as follows: • If is the left inverse of , and , then . • If nonempty is injective, construct a left inverse as follows: for all , if is in the image of , then there exists such that . Let ; this definition is unique because is injective. Otherwise, let be an arbitrary element of .For all , is in the image of . By construction, , the condition for a left inverse. In classical mathematics, every injective function with a nonempty domain necessarily has a left inverse; however, this may fail in
constructive mathematics. For instance, a left inverse of the
inclusion {{math|{0,1} →
R}} of the two-element set in the reals violates
indecomposability by giving a
retraction of the real line to the set {{math|{0,1}}.
Right inverses A
right inverse for (or
section of ) is a function such that : f \circ h = \operatorname{id}_Y . That is, the function satisfies the rule : If \displaystyle h(y) = x, then \displaystyle f(x) = y . Thus, may be any of the elements of that map to under . A function has a right inverse if and only if it is
surjective (this equivalence holds if, and only if, the
axiom of choice holds). : If is the right inverse of , then is surjective. For all y \in Y, there is x = h(y) such that f(x) = f(h(y)) = y. : If is surjective, has a right inverse , which can be constructed as follows: for all y \in Y, there is at least one x \in X such that f(x) = y (because is surjective), so we choose one to be the value of .
Two-sided inverses An inverse that is both a left and right inverse (a
two-sided inverse), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be called
the inverse. : If g is a left inverse and h a right inverse of f, for all y \in Y, g(y) = g(f(h(y)) = h(y). A function has a two-sided inverse if and only if it is bijective. : A bijective function is injective, so it has a left inverse (if is the empty function, f \colon \varnothing \to \varnothing is its own left inverse). is surjective, so it has a right inverse. By the above, the left and right inverse are the same. : If has a two-sided inverse , then is a left inverse and right inverse of , so is injective and surjective.
Preimages If is any function (not necessarily invertible), the
preimage (or
inverse image) of an element is defined to be the set of all elements of that map to : : f^{-1}(y) = \left\{ x\in X : f(x) = y \right\} . The preimage of can be thought of as the
image of under the (multivalued) full inverse of the function . The notion can be generalized to subsets of the range. Specifically, if is any
subset of , the preimage of , denoted by f^{-1}(S) , is the set of all elements of that map to : : f^{-1}(S) = \left\{ x\in X : f(x) \in S \right\} . For example, take the function . This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g. : f^{-1}(\left\{1,4,9,16\right\}) = \left\{-4,-3,-2,-1,1,2,3,4\right\}. The original notion and its generalization are related by the identity f^{-1}(y) = f^{-1}(\{y\}), The preimage of a single element – a
singleton set {{math|{
y} }} – is sometimes called the
fiber of . When is the set of real numbers, it is common to refer to {{math|
f −1({
y})}} as a
level set. ==See also==