Functions and Graphs
The graph of a function is useful if we are trying to model a real-world problem. At times we may not know an expression for a function but we do know some values. The graph can give us a good idea of what function can be applied to the situation to solve the problem.
In mathematics, graph of a function f is the collection of all the ordered pairs (x, f(x)).Particularly, 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 at times referred to as curve sketching. If the function input x is an ordered pair (x1, x2) of real numbers, the graph is collection of ordered triples (x1, x2, f(x1, x2)), and the graphical representation of it is a surface
A 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 by using the Cartesian coordinate system. This system made a lot of new mathematics possible, counting calculus.
The concept of the graph of a function can be generalized to the graph of a relation. Although a function is always identified with the graph of it, they are not the same because it will happen that two functions with different co domain could have the same graph. For instance, the cubic polynomial which is mentioned below is a surjection if the co domain is the real numbers but it is not if its co domain is complex field.
To test if the graph of a curve is a function, use vertical line test. To test if the function is one-to-one, means that it has an inverse function, use the horizontal line test. If the function has an inverse, the graph of inverse can be found by reflecting the graph of original function over the line y = x. A curve is a one-to-one function if it is a function and it passes the horizontal line test. The graph of a function f is set of all the points in the plane of the form (x, f(x)). We could define the graph of f to be the graph of the equation y = f(x).
The graph of a function on real numbers is similar 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 requirement of mathematical statements, for example, the closed graph theorem in functional analysis.
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