Algebra Know-It-ALL

234 The Cartesian Plane

Note the difference in “magnification” between the t and u axes. This difference makes the graph

fit nicely into the available space. Even though an increment on the u axis represents 10 times the

numerical change as an increment of the same length on the t axis, both axes are linear.

Are you confused?

In Chap. 13, you learned that a function never maps a single value of the independent variable to more

than one value of the dependent variable. You can use this fact to determine whether or not a given rela-

tion is a function by looking at its graph. Draw a vertical line somewhere on the graph. “Vertical” in this

context means “parallel to the dependent-variable axis.” Imagine moving this vertical line to the right and

left. Sometimes—maybe all the time—this vertical line will intersect the graph of the relation. For the

relation to qualify as a function, the movable vertical line must never intersect the graph at more than one

point. (It’s okay if there are places, or even large regions, where the vertical line doesn’t intersect the graph

at all.) This trick can be called the vertical-line test.

Here’s a challenge!

How can you tell, merely by looking at their graphs, which of the three relations in this section have

inverses that are functions? Don’t actually graph the inverses. You’ll get a chance to do that in the last three

Practice Exercises

Solution

You can conduct a horizontal-line test on the graph of a relation to see if its inverse is a function. Draw

a horizontal line parallel to the independent-variable axis. Imagine moving this horizontal line up and

down. For the inverse to qualify as a function, this movable line must never intersect the graph of the

original relation at more than one point. (It’s okay if there are places or regions where the horizontal line

doesn’t intersect the graph at all.) Now conduct this test on the graphs shown in Figs. 14-11, 14-12, and

14-13. You’ll see that the line in Fig. 14-11 checks out, so the inverse of this relation is a function. The

same is true for the relation graphed in Fig. 14-13. But the curve in Fig. 14-12 fails the horizontal-line test!

That means that the inverse of that relation is not a function.

Practice Exercises

This is an open-book quiz. You may (and should) refer to the text as you solve these problems.

Don’t hurry! You’ll find worked-out answers in App. B. The solutions in the appendix may not

represent the only way a problem can be figured out. If you think you can solve a particular

problem in a quicker or better way than you see there, by all means try it!

1. Imagine an ordered pair (x,y), and suppose we have plotted its point on the Cartesian
   plane. Neither x nor y is equal to 0, so the point does not fall on either axis. What
   happens to the location of the point if we multiply x by −1 and leave y the same?


2. Imagine an ordered pair (x,y), and suppose we have plotted its point on the
   Cartesian plane. Neither x nor y is equal to 0, so the point does not fall on either axis.
   What happens to the location of the point if we multiply y by −1 and leave
   x the same?