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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.
A ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).
The modified Struve functions L α (x) are equal to −ie −iαπ / 2 H α (ix) and are solutions y(x) ... Graphs, and Mathematical Tables. Applied Mathematics ...
A Cartesian product of two graphs. In graph theory, the Cartesian product G H of graphs G and H is a graph such that: the vertex set of G H is the Cartesian product V(G) × V(H); and; two vertices (u,v) and (u' ,v' ) are adjacent in G H if and only if either u = u' and v is adjacent to v' in H, or; v = v' and u is adjacent to u' in G.
Graphs of the inverse hyperbolic functions The hyperbolic functions sinh, cosh, and tanh with respect to a unit hyperbola are analogous to circular functions sin, cos, tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure.
H(0) = 1 / 2 is often used since the graph then has rotational symmetry; put another way, H − 1 / 2 is then an odd function. In this case the following relation with the sign function holds for all x: () = (+ ). Also, H(x) + H(-x) = 1 for all x.
The coefficient a controls the degree of curvature of the graph; a larger magnitude of a gives the graph a more closed (sharply curved) appearance. The coefficients b and a together control the location of the axis of symmetry of the parabola (also the x-coordinate of the vertex and the h parameter in the vertex form) which is at
Then the Hajós construction forms a new graph that combines the two graphs by identifying vertices v and x into a single vertex, removing the two edges vw and xy, and adding a new edge wy. For example, let G and H each be a complete graph K 4 on four vertices; because of the symmetry of these graphs, the choice of which edge to select from ...