Relations Vs. Functions
All of the relations graphed in Fig. 13 have something in common. For every abscissa, each relation contains at most one ordinate. Never does a curve portray two or more ordinates for a single abscissa, although one of them (the parabola, curve D) has two abscissas for all positive ordinates.
A mathematical relation in which every abscissa corresponds to at most one ordinate is called a function . All of the curves shown in Fig. 13 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 (if we want to “go nonstandard” and consider y as the independent variable and x as the dependent variable).
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).
The Cartesian Plane Practice Problems
Practice 1
Suppose a certain relation has a graph that looks like a circle. Is this a function of y in terms of x ? Is it a function of x in terms of y?
Solution 1
The answer is no in both cases. Figure 14 shows why. A simple visual “test” to determine whether or not a given relation is a function is to imagine an infinitely long, straight line parallel to the dependentvariable axis, and that can be moved back and forth. If the curve ever intersects the line at more than one point, then the curve is not a function.
A “vertical line” (parallel to the y axis) test can be used to determine whether or not the circle is a function of the form y = f ( x ), meaning “y is a function of x .” Obviously, the answer is no, because there are some positions of the line for which the line intersects the circle at two points.
A “horizontal line” (parallel to the x axis) test can be used to determine if the circle is a function of the form x = f ( y ), meaning “ x is a function of y.” Again the answer is no; there are some positions of the line for which the line intersects the circle twice.
Practice 2
How could the circle as shown in Fig. 14 be modified to become a function of y in terms of x ?
Solution 2
Part of the circle must be removed, such that the resulting curve passes the “vertical line” test. For example, either the upper or the lower semicircle can be taken away, and the resulting graph will denote y as a function of x . But these are not the only ways to modify the circle to get a graph of a function. There are infinitely many ways in which the circle can be partially removed or broken up in order to get a graph of a function. Use your imagination!
Practice Problems for these concepts can be found at: The Circle Model Practice Test
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