determined. There are computer programs that generate detailed graphs of relations by plotting millions
of points and connecting them by means of a scheme called curve fitting.
Here’s a challenge!
Draw graphs of the inverses of the three relations in this section.
Solution
First, let’s figure out the equations for these inverse relations. To do that, we must transpose the values of
the variables without changing their names. We must also transpose the domain and the range. The three
original relations are
y=x+ 2
y=±(x1/2)
y=x1/2
In Chap. 13, we were given the domains and ranges for these relations. In the first case, the domain and
range are both the entire set of reals. In the second case, the domain is the set of nonnegative reals, and the
range is the set of all reals. In the third case, the domain and range are both the set of nonnegative reals.
Switching the values of the variables by reversing their positions in the original equation, we get
x=y+ 2
x=±(y1/2)
x=y1/2
When we manipulate these equations to get x in terms of y, the results are
y=x− 2
y=x^2
y= (+x)^2
When we transpose the domain and the range from the original relations in the first case, they both remain
the entire set of reals. In the second case, the domain becomes the set of all reals, and the range becomes
the set of nonnegative reals. In the third case, the domain and range both remain the set of nonnegative
reals. The plus sign in the last equation means that we consider it only for nonnegative values of x, because
negative values of x aren’t part of the domain.
Now that we know the equations for the inverse relations, we’re ready to graph them. A simple trick
makes it easy to graph the inverse of any relation. We draw the line representing all points where the inde-
pendent and dependent variables have the same value. We can imagine this line, represented by the equation
y=x in these examples, as a “point reflector.” It works like the “number reflector” for generating negative
numbers from positive numbers on the number line. (We devised that gimmick in Chap. 3. Now we’re
operating in two dimensions instead of one.) For any point that’s part of the graph of the original relation,
we can find its counterpart in the graph of the inverse relation by going to the opposite side of the “point
reflector,” exactly the same distance away. Figure 14-7 shows how this works. The line connecting a point in
the original graph and its “mate” in the inverse graph is perpendicular to the “point reflector.” In addition,
the “point reflector” intersects every point-connecting line exactly in the middle. Technically, we say that the
“point reflector” is a perpendicular bisector of any line connecting a point with its inverse point.
Three Relations 229