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The notation might be used for the inverse function to avoid ambiguity with the multiplicative inverse. [ 7 ] In keeping with the general notation, some English authors use expressions like sin −1 ( x ) to denote the inverse of the sine function applied to x (actually a partial inverse ; see below).
Assuming that has an inverse in a neighbourhood of and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at and have a derivative given by the above formula. The inverse function rule may also be expressed in Leibniz's notation. As that notation suggests,
Although many functions do not have an inverse, every relation does have a unique converse. The unary operation that maps a relation to the converse relation is an involution , so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on the category of relations ...
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative, 1 1 − z 2 {\textstyle {\tfrac {1}{\sqrt {1-z^{2}}}}} , as a binomial series , and integrating term by term (using the integral definition as ...
However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan −1 = arctan ≠ 1/tan. In some cases, when, for a given function f, the equation g ∘ g = f has a unique solution g, that function can be defined as the functional square root of f, then written as g = f 1/2.
The notation f −1 is sometimes also used for the inverse function of the function f, which is for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse 1/(sin x) = (sin x) −1 is the cosecant of x, and not the inverse sine of x denoted by sin −1 x or arcsin x.
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
For a function to have an inverse, it must be one-to-one.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 = defined on the whole of is not one-to-one since = for any .