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Is it possible for a relation to be its own inverse
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Answer 12-4
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Yes The simplest example is the relation described in the Cartesian xy plane by the equation y=x which can also be written as f (x) = x Another, less obvious example, is y = x
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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 20-2
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which can also be written as f (x) = x If a relation s graph is a circle centered at the origin, then that relation is its own inverse Examples include all relations of the form x2 + y2 = r2 where r is the radius of the circle We can also write such a relation in the form f (x) = (r2 x2)1/2
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Question 12-5
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How can we tell, simply by looking at the graph of a relation, whether or not that relation is its own inverse
Answer 12-5
Suppose that when we flip the graph over in three dimensions along the line y = x as if that line were the hinge of a revolving door, we end up with exactly the same graph as the one we started with In any case like that, the relation is its own inverse If we do the revolving door transformation and end up with a graph that s different in any way from the one we started
Part Two 405
y Graph of inverse relation 6 4 2 x 6 y=x 4 2 2 4 6 Graph of original relation 2 4 6
Flip whole graph over by 1/2 revolution along this line
Figure 20-3
Illustration for Question and Answer 12-5
with, then the relation isn t its own inverse Figure 20-3 shows an example of a graph of the second type, where the inverse obviously differs from the original relation
Question 12-6
Suppose we have a relation that s not a function, because it maps some values of the independent variable x to more than one value of the dependent variable y Is it possible to modify such a relation so that it becomes a function of x
Answer 12-6
Yes, in most cases it s possible If we can restrict the domain or the range to values such that the modified relation never maps any value of x to more than one value of y, then the modified relation is a function of x
Question 12-7
Is the inverse of a function always a function
Answer 12-7
No, not always Suppose we have a function f that maps values of an independent variable x to values of a dependent variable y Also imagine that, for any value of x in the domain, there s
Review Questions and Answers
only one corresponding value of y in the range On that basis, we know that f is a function of x However, if some values of y are mapped from two or more values of x, then we don t have a function of y when we consider y as the independent variable and x as the dependent variable Although the inverse f 1 is a relation, it s not a true function
Question 12-8
Consider a function f that maps values of x to values of y Suppose that f 1, which maps values of y to values of x, is a relation but not a true function Is it possible to modify the inverse relation f 1 so that it becomes a function of y
Answer 12-8
In most cases, yes If we can restrict the inverse relation s domain (the set of y values for which f 1 is defined) or the inverse relation s range (the set of x values for which f 1 is defined) so that the modified version of f 1 never maps any value of y to more than one value of x, then the modified inverse is a true function of y
Question 12-9
Consider the two functions f (x) = x and g (x) = x Both f and g are their own inverses, and the inverses are also true functions Is it possible for any other true function to be its own inverse, with that inverse also constituting a true function
Answer 12-9
Yes, this can happen Consider the real-number function h (x) = 1/x where x 0 This function is its own inverse, because h 1[h (x)] = h 1(1/x) = 1/(1/x) = x and h[h 1(x)] = h (1/x) = 1/(1/x) = x
Question 12-10
Imagine a function f such that y = f (x)
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