Graph of a function

As a function graph or shortly graph ( rare: Function Graf or Count ) of a function is called in mathematics the set of all ordered pairs of elements of the set of definitions and the corresponding function values ​​.

Sometimes these pairs can be interpreted as points in the plane or in the space of intuition, they will also bend, curve or function is also called a graph.

Formal definition

The graph of a function with the set of definitions and the target amount is the amount

Special cases and examples

The graph of a function is a subset of, and can therefore be regarded as a point quantity or geometric figure in the plane.

For example, the following applies:

  • Linear functions have a graph is a straight line.
  • The graph of the quadratic function is a parabola.
  • The graph of the reciprocal function ( with ) is a hyperbola.

The graphs of functions or are subsets of and can be represented pictorially as yet spatial figures also.

With a sufficiently smooth functions, the graph is a surface in three dimensional space. For example, the graph of the function results in an elliptic paraboloid.

Use in mathematics

In set-theoretic definitions of functions, these are often just defined as a set of point - value pairs, ie, the graph would be nothing more than the function itself In mathematical considerations that are not directly in the context of set-theoretic foundation of mathematical concepts, is given to However, usually no set structure of a function requires, but demands only the definedness of the image at a given location. Set operations are then not executed features (such as would usually not as meaningful expression seen ), but in some cases it is just practically be regarded as a set with the operations and properties defined on sets a function; this analysis is done on the graph of the function. Besides the possibility to consider a function thereby as a geometric figure, are mentioned here as further examples:

  • In every Polish space is a function if and only Borel - measurable function if the graph is a Borel set.
  • Set by the closed graph: A linear operator between Banach spaces is continuous if its graph is complete.

Graph in accordance with the graphical representation

The graph is a mathematical object. They serve as part of the mathematics of Veranschauung and let assumptions about the properties of a function.

Graph discontinuous functions, defining gaps

In the representation of the graphs of discontinuous functions or functions with gaps definition is often indicated by that point to the graph belongs, and that a point is not part of the graph. An example is an illustration of the signum function.

Examples

Three examples of graphs of functions:

Euclidean space Plane (geometry) Line (geometry) Closed graph theorem American Mathematical Society
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