**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.

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.

**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.

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