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At A, Cartesian graph of the relation y = x1/2 At B, Cartesian graph of the inverse relation
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It s reasonable for you to wonder, Can any relation be its own inverse The answer is yes There are plenty of examples Consider the following equation: x2 + y2 = 25 The Cartesian graph of this equation is a circle centered at the origin and having a radius of 5 units (Fig 12-5) If we transpose the variables, we get y2 + x2 = 25 which is equivalent to the original relation If we perform the graphical transformation by mirroring the circle around the line y = x, we get another circle having the same radius and the same center Theoretically, all but two of points on the new circle are in different places than the points on the original circle, but the graph looks the same as the one shown in Fig 12-5
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Here s a challenge!
Consider the following relation where the independent variable is x and the dependent variable is y: x 2/9 + y2/25 = 1
y Circle centered at the origin is symmetrical 6 4 2
Point reflector line
x 6 4 2 2 4 6 with respect to the point reflector line 2 4 6
Figure 12-5 Cartesian graph of the relation x2 + y2 = 25
This relation is its own inverse
Finding an Inverse Relation
y 6 4 2 x 6 4 2 2 4 6 2 4 6
Ellipse centered at origin
Figure 12-6
Cartesian graph of the relation x2/9 + y2/25 = 1
Figure 12-6 is a graph of this relation in Cartesian coordinates It s an ellipse centered at the origin The distance from the center to the extreme right- or left-hand point on the ellipse measures 3 units, which is the square root of 9 The distance from the center to the uppermost or lowermost point on the ellipse measures 5 units, which is the square root of 25 Determine the inverse of this relation, and graph it
Solution
We can obtain the inverse of this relation by swapping the variables That gives us the equation y 2/9 + x 2/25 = 1 which can be rewritten as x 2/25 + y 2/9 = 1 Figure 12-7 illustrates the graphs of the original relation and its inverse in Cartesian coordinates The new graph is another ellipse having the same shape as the original one, and centered at the origin just like the original one But the horizontal and vertical axes of the ellipse have been transposed The distance from the center to the extreme right- or left-hand point on the inverse ellipse measures 5 units, which is the square root of 25 The distance from the center to the uppermost or lowermost point on the inverse ellipse measures 3 units, which is the square root of 9
Inverse Relations in Two-Space
y Original relation 6 4 2
Point reflector line
x 6 4 2 2 4 6 Inverse relation 2 4 6
Figure 12-7
Cartesian graph of the relation x2/25 + y2/9 = 1, the inverse of the relation graphed in Fig 12-6
Finding an Inverse Function
If a function is a bijection (that is, a perfect one-to-one correspondence) over a certain domain and range, then we can transpose the domain and range, and the resulting inverse relation will always be a function If a function is many-to-one, then its inverse relation is one-to-many, so it s not a function
Undoing the work Suppose that f and f 1 are both true functions that are inverses of each other Then for all x in the domain of either function, we have
f 1[ f (x)] = x and f [ f 1(x)] = x An inverse function undoes the work of the original function in an unambiguous manner when the domains and ranges are restricted so that the original function and the inverse are both bijections
Finding an Inverse Function
Sometimes we can simply turn f inside-out to get an inverse relation, and the inverse will be a true function for all the values in the domain and range of f But often, when we seek the inverse of a function f, we get a relation that s not a true function, because some elements in the range of f map from more than one element in the domain of f When this happens, we must restrict f to define an inverse f 1 that s a true function We can usually (but not always) find a way to force f 1 to behave as a true function by excluding all values of either variable that map to more than one value of the other variable Once we ve done that, we get a bijection, ensuring that there is no ambiguity or redundancy either way
Making a relation behave as a function A little while ago, we looked at a relation whose graph is a circle with a radius of 5 units (Fig 12-5) The equation of that relation, once again, is
x 2 + y 2 = 25 which can be rewritten as y 2 = 25 x 2 and then morphed to y = (25 x 2)1/2 If we use relation notation to express this equation and name the relation f, we have f (x) = (25 x 2)1/2 The vertical-line test tells us that f is not a true function of x We can modify it so that it becomes a function of x if we restrict the range to nonnegative values Graphically, that eliminates the lower half of the circle, so that for every input value in the domain, we get only one output in the range Figure 12-8A is an illustration of this function, which we can call f+ and define as f+(x) = (25 x 2)1/2 Once again, we mustn t forget that when we take the 1/2 power of a quantity without including any sign, we mean, by default, the nonnegative square root of that quantity The solid dots indicate that the plotted points are part of the range of f+ Now suppose that we eliminate the top half of the circle including the points ( 5,0) and (5,0), getting the graph shown in Fig 12-8B The vertical-line test indicates that this is a true function of x If we call this function f , we can write f (x) = (25 x2)1/2 The white dots (small open circles) tell us that the plotted points are not part of the range of f We can restrict the range further, say to values strictly larger than 1 or values smaller than or
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