Relation versus Function Help

Introduction to Relations and Functions

It’s important to know the similarities and differences between two concepts as they pertain to coordinate geometry: the idea of a relation and the idea of a function . A relation is an equation or formula that relates the value of one variable to that of another. A function is a relation that meets certain requirements.

Relations

Mathematical relations between two variables x and y are often written in such a way that y is expressed in terms of x . When this is done, y is the dependent variable and x is the independent variable. The following are some examples of relations denoted this way:

y = 5

y = x + 1

y = 2 x

y = x ²

Some Simple Graphs

Figure 6-3 shows how the graphs of the above equations look on the Cartesian plane. Mathematicians and scientists call such a graph a curve , even if it happens to be a straight line.

Fig. 6-3 . Graphs of four relations in Cartesian coordinates. The relations shown by A, B, and C are linear, but the relation shown by D is not.

The graph of y = 5 (curve A) is a horizontal line passing through the point (0,5) on the y axis. The graph of y = x + 1 (curve B) is a straight line that ramps upward at a 45° angle (from left to right) and passes through (0,1) on the y axis. The graph of y = 2 x (curve C) is a straight line that ramps upward more steeply, and that passes through the origin (0,0). The graph of y = x ² (curve D) is known as a parabola . In this case the parabola rests on the origin (0,0), opens upward, and is symmetrical with respect to the y axis.

Relations Vs Functions

All of the relations graphed in Fig. 6-3 have something in common. For every abscissa, each relation contains at most one ordinate. Never does a curve have two or more ordinates for a single abscissa, although one of them (the parabola, curve D) has two abscissas for all positive ordinates.

Fig. 6-3 . Graphs of four relations in Cartesian coordinates. The relations shown by A, B, and C are linear, but the relation shown by D is not.

Functions

A function is a mathematical relation in which every abscissa corresponds to at most one ordinate. According to this criterion, all the curves shown in Fig. 6-3 are graphs of functions of y in terms of x . In addition, curves A, B, and C show functions of x in terms of y . But curve D does not represent a function of x in terms of y . If x is considered the dependent variable, then there are some values of y (that is, some abscissas) for which there exist two values of x (ordinates).

Functions are denoted as italicized letters of the alphabet, usually f, F, g, G, h , or H , followed by the independent variable or variables in parentheses. Examples are:

f ( x ) = x + 1

g ( y ) = 2 y

H ( z ) = z ²

These equations are read “ f of x equals x plus 1,” “ g of y equals 2 y ”, and “ H of z equals z squared,” respectively.

Practice problems for these concepts can be found at: The Cartesian Plane Practice Test.