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A function can do this
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Figure 11-2 A true function never assigns any element in its
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Example 1 revisited Let s take another look at the relation given by Example 1 in the previous section We described it using the following equation:
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y=x 1 Figure 11-3 is a graph of this equation in the Cartesian xy plane It s a straight line with a slope of 1 and a y intercept of 1 If we imagine an infinitely long, movable vertical line sweeping back and forth, it s easy to see that the vertical line never intersects our graph at more than one point Therefore, the relation is a function
Example 2 revisited The relation in Example 2 in the previous section has a graph that s a parabola opening upward, as shown in Fig 11-4 The equation is
y = x2 The vertex of the parabola represents the absolute minimum value of the relation, and it coincides with the coordinate origin (0,0) The curve rises symmetrically on either side of the y axis It s not difficult to see that a movable vertical line never intersects the parabola at more than one point This fact tells us that the relation is a function of x
y 6 4 2 x 6 4 2 2 4 6
4 6
Movable vertical line
Figure 11-3 Cartesian graph of the relation y = x 1
The vertical-line test reveals that it s a function of x
Relations in Two-Space
y 6 4 2 x 6 4 2 2 4 6 Movable vertical line 2 4 6
Figure 11-4 Cartesian graph of the relation y = x2 The
vertical-line test reveals that it s a function of x
Example 3 revisited Figure 11-5 is a graph of the relation we saw in Example 3 in the previous section The equation for that relation was stated as
y = (x1/2) In this case, the graph is a parabola that opens to the right The vertex coincides with the coordinate origin, but there is no absolute minimum or maximum for the dependent variable When we construct a movable vertical line in this situation, we find that it doesn t intersect the graph when x < 0 When x = 0, the vertical line intersects the graph at the single point (0,0) When x > 0, the vertical line intersects the graph at two points Therefore, this relation is not a function of x
Example 4 revisited Figure 11-6 is a graph of the relation we saw in Example 4 in the previous section It s the upper half of the parabola of Fig 11-5, with the point (0,0) included The equation is
y = x1/2 The vertical-line test tells us that this relation is a function of x No matter where we position the vertical line, it never intersects the graph more than once
What s a Two-Space Function
y 6 4 2 x 6 4 2 2 4 6 Movable vertical line 2 4 6
Figure 11-5 Cartesian graph of the relation y = (x1/2)
The vertical-line test reveals that it isn t a function of x
y 6 4 2 x 6 4 2 2 4 6 Movable vertical line 2 4 6
Figure 11-6 Cartesian graph of the relation y = x1/2
The vertical-line test reveals that it s a function of x
Relations in Two-Space
Are you confused
By now you might wonder, When we have a relation where the independent variable is represented by the polar angle q and the dependent variable is represented by the polar radius r, how can we tell if the relation is a function of q It s easy, but there s a little trick involved We can draw the graph of the relation in a Cartesian plane with q on the horizontal axis and r on the vertical axis We must allow both q and r to attain all possible real-number values Once we ve drawn the graph of the polar relation the Cartesian way, we can use the Cartesian vertical-line test to see whether or not the relation is a function of q
Here s a challenge!
Consider the relation between an independent variable x and a dependent variable y such that x2 y2 = 1 Sketch a graph of this relation in the Cartesian xy plane Use the vertical-line test to determine, on the basis of the graph, whether or not this relation is a function of x
Solution
Figure 11-7 is a graph of this relation It s a geometric figure called a hyperbola The vertical-line test tells us that the relation is not a function of x
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