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In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective , and if it exists, is denoted by f − 1 . {\displaystyle f^{-1}.}
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
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 functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
A ray through the unit hyperbola = in the point (,), where is twice the area between the ray, the hyperbola, and the -axis. The earliest and most widely adopted symbols use the prefix arc-(that is: arcsinh, arccosh, arctanh, arcsech, arccsch, arccoth), by analogy with the inverse circular functions (arcsin, etc.).
The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution). Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25).
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 ...
Pages in category "Inverse functions" The following 17 pages are in this category, out of 17 total. This list may not reflect recent changes. ...
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