Search results
Results from the WOW.Com Content Network
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.
Given its domain and its codomain, a function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function, a popular means of illustrating the function. [note 1] [4] When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.
For example, if is a function of and , then its partial derivatives measure the variation in in the and direction. However, they do not directly measure the variation of f {\displaystyle f} in any other direction, such as along the diagonal line y = x {\displaystyle y=x} .
A function f from X to Y. The set of points in the red oval X is the domain of f. Graph of the real-valued square root function, f(x) = √ x, whose domain consists of all nonnegative real numbers. In mathematics, the domain of a function is the set of inputs accepted by the function.
A differential field F is a field F 0 (rational functions over the rationals Q for example) together with a derivation map u → ∂u. (Here ∂u is a new function. Sometimes the notation u′ is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear
For the depicted f, a, and b, we can ensure that the value f(x) is within an arbitrarily small interval (b – ε, b + ε) by restricting x to a sufficiently small interval (a – δ, a + δ). Hence f(x) → b as x → a.
To avoid ambiguity, some mathematicians [citation needed] choose to use ∘ to denote the compositional meaning, writing f ∘n (x) for the n-th iterate of the function f(x), as in, for example, f ∘3 (x) meaning f(f(f(x))). For the same purpose, f [n] (x) was used by Benjamin Peirce [14] [11] whereas Alfred Pringsheim and Jules Molk suggested ...
For example, it may represent an exponential function when its values are expressed in the logarithmic scale. It means that when log(g(x)) is a linear function of x, the function g is exponential. With linear functions, increasing the input by one unit causes the output to increase by a fixed amount, which is the slope of the graph of the function.