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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.
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.
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. It is sometimes denoted by or , where f is the function. In layman's terms, the domain of a function can generally be thought of as ...
A real function f is even if, for every x in its domain, −x is also in its domain and [1]: p. 11 = or equivalently () = Geometrically, the graph of an even function is symmetric with respect to the y -axis, meaning that its graph remains unchanged after reflection about the y -axis.
In mathematics, the hypograph or subgraph of a function. {\displaystyle f:\mathbb {R} ^ {n}\rightarrow \mathbb {R} } is the set of points lying on or below its graph. A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph. The domain (rather than the codomain) of the function is ...
Linear function (calculus) In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane. [ 1] The characteristic property of linear functions is that when the input variable is changed, the change in the output ...
Closed graph property. In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. [1] [2] A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.
Not all functions have fixed points: for example f(x) = x + 1 has no fixed points because x is never equal to x + 1 for any real number. In graphical terms, a fixed-point x of a function of one variable is an x such that the point ( x , f ( x )) is on the line y = x , or in other words the graph of f has a point in common with the graph of the ...