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Finding an Inverse Relation
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Replacing y by g 1(x), we get g 1(x) = x 2 The domain of the original relation g spans the set of nonnegative reals, and the range of g spans the set of all reals Therefore, the domain of g 1 includes all reals, while the range of g 1 is confined to the set of nonnegative reals
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Still another example Let s find the inverse of the relation
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h(x) = x 1/2 When we write the 1/2 power of a quantity without including any sign, we mean the nonnegative square root of that quantity If we call the dependent variable y, then we have y = x1/2 Swapping the names of the variables, we get x = y1/2 Squaring both sides, we obtain x2 = y Reversing the left- and right-hand sides of this equation yields y = x2 Replacing y by h 1(x), we get h 1(x) = x 2 Is the inverse of h identical to the inverse of g we obtained a few moments ago It looks that way on the surface, but it s not so simple when we examine the situation more closely The domain of h spans the set of nonnegative reals, just as the domain of g does But the range of h spans the set of nonnegative reals only (not the set of all reals, as the range of g does) Transposing, we must conclude that the domain and range of h 1 are both confined to the set of nonnegative reals The relations h and h 1 are therefore restricted versions of g and g 1
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The graphical way Imagine the line represented by the equation y = x in the Cartesian xy plane as a point reflector For any point that s part of the graph of the original relation, we can locate its
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Inverse Relations in Two-Space
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y 6 4 Inverse Original 2 y=x Original Point reflector line Inverse
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x 6 Inverse 4 2 2 Original 4 6 Original Inverse 2 4 6
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Figure 12-1
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Any point on the graph of the inverse of a relation is the point s image on the opposite side of a point reflector line The new coordinates are obtained by reversing the sequence of the ordered pair representing the original point
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counterpart in the graph of the inverse relation by going to the opposite side of the point reflector, exactly the same distance away Figure 12-1 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 The point reflector is a perpendicular bisector of every point-connecting line Mathematically, we can do a point transformation of the sort shown in Fig 12-1 by reversing the sequence of the ordered pair representing the point For example, if (4,6) represents a point on the graph of a certain relation, then its counterpoint on the graph of the inverse relation is represented by (6,4) When we want to graph the inverse of a relation, we flip the whole graph over along a hinge corresponding to the point reflector line y = x That moves every point in the graph of the original relation to its new position in the graph of the inverse Figures 12-2, 12-3, and 12-4 show how this process works with the three relations we dealt with a few moments ago The positions of the x and y axes haven t changed, but the values of the variables, as well as the domain and range, have been reversed
Finding an Inverse Relation
y 6 4 2 (0, 2) x 6 ( 5, 3) 4 6 4 2 2 2 4 6 (4, 6)
y 6 (6, 4) 4 2 (2, 0)
6 4 2 2 4 ( 3, 5) 2 4 6
Figure 12-2
At A, Cartesian graph of the relation y = x + 2 At B, Cartesian graph of the inverse relation
Inverse Relations in Two-Space
y 6 (1, 1) 4 (0, 0) 2 (4, 2)
6 4 2 2 (4, 2) 4 (1, 1) 6 2 6
y 6 ( 2, 4) ( 1, 1) 2 (2, 4) (1, 1) x 6 4 (0, 0) 4 6 2 2 2 4 6
Figure 12-3