Functions and Graphs

A very significant development in mathematics was the introduction of the Cartesian Coordinate system (or x-y coordinate system),. We usually draw the graph of a function using the Cartesian coordinate system. This system made a lot of new mathematics possible, including calculus.

The graph of a function is really useful if we are trying to model a real-world problem. Sometimes we may not know an expression for a function but we do know some values (maybe from an experiment). The graph can give us a good idea of what function may be applied to the situation to solve the problem.

In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)). In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching. If the function input x is an ordered pair (x1, x2) of real numbers, the graph is the collection of all ordered triples (x1, x2, f(x1, x2)), and its graphical representation is a surface

The graph of a function on real numbers is identical to the graphic representation of the function. For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e.g., the closed graph theorem in functional analysis.

The concept of the graph of a function is generalized to the graph of a relation. Note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different co domain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its co domain is the real numbers but it is not if its co domain is the complex field.

To test if a graph of a curve is a function, use the vertical line test. To test if the function is one to one, meaning it has an inverse function, use the horizontal line test. If the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function over the line <y = x. A curve is a one to one function if and only if it is a function and it passes the horizontal line test. The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). So, the graph of a function if a special case of the graph of an equation.