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The function g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image. A function f with nonempty domain is injective if and only if it has a left inverse. [21] An elementary proof runs as follows: If g is the left inverse of f, and f(x) = f(y), then g(f(x)) = g(f(y)) = x = y.
In number theory, reversing the digits of a number n sometimes produces another number m that is divisible by n. This happens trivially when n is a palindromic number; the nontrivial reverse divisors are 1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ... (sequence A008919 in the OEIS).
The reciprocal function: y = 1/x.For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.. In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1.
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 : ′ = ′ = ′ (()).
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 ...
Trapdoor functions are a special case of one-way functions and are widely used in public-key cryptography. [2] In mathematical terms, if f is a trapdoor function, then there exists some secret information t, such that given f(x) and t, it is easy to compute x. Consider a padlock and its key. It is trivial to change the padlock from open to ...
[36] [37] The connection is made through the Busy Beaver function, where BB(n) is the maximum number of steps taken by any n state Turing machine that halts. There is a 15 state Turing machine that halts if and only if a conjecture by Paul Erdős (closely related to the Collatz conjecture) is false.
Reverse accumulation traverses the chain rule from outside to inside, or in the case of the computational graph in Figure 3, from top to bottom. The example function is scalar-valued, and thus there is only one seed for the derivative computation, and only one sweep of the computational graph is needed to calculate the (two-component) gradient.