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The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x 2 + y 2.Around point A, y can be expressed as a function y(x).In this example this function can be written explicitly as () =; in many cases no such explicit expression exists, but one can still refer to the implicit function y(x).
Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation y = f ( x ) {\displaystyle y=f(x)} in which the functional form is explicitly stated; this is called an explicit representation.
Other specific graphs that are unit distance graphs include the Petersen graph, [10] the Heawood graph, [11] the wheel graph (the only wheel graph that is a unit distance graph), [3] and the Moser spindle and Golomb graph (small 4-chromatic unit distance graphs). [12] All generalized Petersen graphs, such as the Möbius–Kantor graph depicted ...
By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval [−2, 2] and the image is [−1, 1]; for the second one, the domain is [−2, ∞) and the image is [1, ∞); for the last one, the domain is (−∞, 2] and the image is (−∞, −1]. As the three graphs together form a ...
Defining g −1 as the inverse of g is an implicit definition. For some functions g, g −1 (y) can be written out explicitly as a closed-form expression — for instance, if g(x) = 2x − 1, then g −1 (y) = 1 / 2 (y + 1). However, this is often not possible, or only by introducing a new notation (as in the product log example below).
The number of perfect matchings of the complete graph K n (with n even) is given by the double factorial (n – 1)!!. [12] The crossing numbers up to K 27 are known, with K 28 requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project. [13] Rectilinear Crossing numbers for K n are
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.