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In mathematics, the **graph** of a function *f* is the set of ordered pairs (*x*, *y*), where *f*(*x*) = *y*. In the common case where x and *f*(*x*) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane.

- Definition
- Examples
- Functions of one variable
- Functions of two variables
- Generalizations
- See also
- References
- External links

In the case of functions of two variables, that is functions whose domain consists of pairs (*x*, *y*), the graph usually refers to the set of ordered triples (*x*, *y*, *z*) where *f*(*x*, *y*) = *z*, instead of the pairs ((*x*, *y*), *z*) as in the definition above. This set is a subset of three-dimensional space; for a continuous real-valued function of two real variables, it is a surface.

A graph of a function is a special case of a relation.

In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using rectangular axes; see * Plot (graphics) * for details.

In the modern foundations of mathematics, and, typically, in set theory, a function is actually equal to its graph.^{ [1] } However, it is often useful to see functions as mappings,^{ [2] } which consist not only of the relation between input and output, but also which set is the domain, and which set is the codomain. For example, to say that a function is onto (surjective) or not the codomain should be taken into account. The graph of a function on its own doesn't determine the codomain. It is common^{ [3] } to use both terms *function* and *graph of a function* since even if considered the same object, they indicate viewing it from a different perspective.

Given a mapping , in other words a function together with its domain and codomain , the graph of the mapping is^{ [4] } the set

- ,

which is a subset of . In the abstract definition of a function, is actually equal to .

One can observe that, if, , then the graph is a subset of (strictly speaking it is , but one can embed it with the natural isomorphism).

The graph of the function defined by

is the subset of the set

From the graph, the domain is recovered as the set of first component of each pair in the graph . Similarly, the range can be recovered as . The codomain , however, cannot be determined from the graph alone.

The graph of the cubic polynomial on the real line

is

If this set is plotted on a Cartesian plane, the result is a curve (see figure).

The graph of the trigonometric function

is

If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see figure).

Oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the function surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function:

The graph of a function is contained in a Cartesian product of sets. An X–Y plane is a cartesian product of two lines, called X and Y, while a cylinder is a cartesian product of a line and a circle, whose height, radius, and angle assign precise locations of the points. Fibre bundles are not Cartesian products, but appear to be up close. There is a corresponding notion of a graph on a fibre bundle called a section.

A **complex number** is a number that can be expressed in the form *a* + *b i*, where a and b are real numbers, and i represents the “*imaginary unit*”, satisfying the equation Because no real number satisfies this equation, i is called an imaginary number. For the complex number *a* + *b i*, a is called the **real part** and b is called the **imaginary part**. The set of complex numbers is denoted by either of the symbols ℂ or **C**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

In mathematics, a **continuous function** is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be *discontinuous*. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.

A **Cartesian coordinate system** in a plane is a coordinate system that specifies each point uniquely by a pair of numerical **coordinates**, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a *coordinate axis* or just *axis* of the system, and the point where they meet is its *origin*, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

In mathematics, one can often define a **direct product** of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.

In mathematics, an **exponential function** is a function of the form

In mathematics, an **inverse function** is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, i.e., *g*(*y*) = *x* if and only if *f*(*x*) = *y*. The inverse function of f is also denoted as .

In mathematics, the **domain** or **set of departure** of a function is the set into which all of the input of the function is constrained to fall. It is the set X in the notation *f*: *X* → *Y*, and is alternatively denoted as . Since a function is defined on its entire domain, its domain coincides with its domain of definition. However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain.

In mathematics, the **codomain** or **set of destination** of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation *f*: *X* → *Y*. The term range is sometimes ambiguously used to refer to either the codomain or image of a function.

In mathematics, a **function** is a binary relation between two sets that associates every element of the first set to exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In mathematics, the **range of a function** may refer to either of two closely related concepts:

In calculus, a **differentiable function** of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth and does not contain any break, angle, or cusp.

In mathematics, more specifically in multivariable calculus, the **implicit function theorem** is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a **function of a real variable** is a function whose domain is the real numbers ℝ, or a subset of ℝ that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the **real functions**, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

In mathematical analysis, the **smoothness** of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered "smooth" if it is differentiable everywhere. At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be **infinitely differentiable** and referred to as a **C-infinity function**.

In mathematics, the **restriction** of a function is a new function, denoted or , obtained by choosing a smaller domain *A* for the original function .

In mathematics, the **epigraph** or **supergraph** of a function is the set of points lying on or above its graph.

In mathematics, the **hypograph** or **subgraph** of a function *f* : **R**^{n} → **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.

In mathematics, specifically set theory, the **Cartesian product** of two sets *A* and *B*, denoted *A* × *B*, is the set of all ordered pairs (*a*, *b*) where *a* is in *A* and *b* is in *B*. In terms of set-builder notation, that is

In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a **function of several real variables** or **real multivariate function** is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article.

- ↑ Charles C Pinter (2014) [1971].
*A Book of Set Theory*. Dover Publications. p. 49. ISBN 978-0-486-79549-2. - ↑ T. M. Apostol (1981).
*Mathematical Analysis*. Addison-Wesley. p. 35. - ↑ P. R. Halmos (1982).
*A Hilbert Space Problem Book*. Springer-Verlag. p. 31. ISBN 0-387-90685-1. - ↑ D. S. Bridges (1991).
*Foundations of Real and Abstract Analysis*. Springer. p. 285. ISBN 0-387-98239-6.

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- Weisstein, Eric W. "Function Graph." From MathWorld—A Wolfram Web Resource.

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