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Figure 11-10 Illustration for Problem 1 2 Imagine a relation in which the domain X is the set of all positive rational numbers, while the range Y is the set of all positive integers Let s call the independent variable x and the dependent variable y Suppose that for any x in set X, the relation rounds x up to the next larger integer to obtain the corresponding element y in set Y Is this relation an injection Is it a surjection Is it a bijection Is it a function of x Explain each answer 3 Suppose that we reverse the action of the relation described in Problem 2 Let the domain X be the set of all positive integers, while the range Y is the set of all positive rationals Suppose that for any value of the independent variable x in set X, the relation
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maps to the set of all rationals in the half open interval (x 1,x] Is this relation an injection Is it a surjection Is it a bijection Is it a function of x Explain each answer 4 Can a relation whose graph is a circle or ellipse in the Cartesian xy plane ever be a function of x Why or why not 5 Can a relation whose graph is a circle in the polar qr plane ever be a function of q Why or why not 6 Find all the sums, differences, products, and ratios of f (x) = x + 2 and g (x) = 3 7 Find all the sums, differences, products, and ratios of f (x) = x + 1 and g (x) = x 1 8 Find all the sums, differences, products, and ratios of f (x) = x 1 and g (x) = x 2 9 Find all the sums, differences, products, and ratios of f (x) = sin2 q and g (x) = cos2 q 10 What are the real-number domains of all the original and derived functions in Problems 6 through 9
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Any relation in two-space has a unique inverse relation, which can be called simply the inverse if we understand that we re dealing with a relation We denote the fact that a relation is an inverse by writing a superscript 1 after its name For example, if we have a relation f (x), then its inverse is f 1(x)
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Finding an Inverse Relation
A relation s inverse does the opposite of whatever the original relation does To find the inverse of a relation, we can manipulate the equation so that the independent and dependent variables switch roles We must therefore transpose the domain and range
The algebraic way Suppose we have a relation f (x) The inverse of f, which we call f 1, is another relation such that
f 1[ f (x)] = x for all possible values of x in the domain of f, and f [ f 1( y)] = y for all possible values of y in the range of f When we talk or write about an inverse relation, it s customary to swap the names of the variables so the inverse relation calls the independent and dependent variables by their original names That means the preceding equation can be rewritten as f [ f 1(x)] = x for all possible values of x in the domain of f 1
Inverse Relations in Two-Space
An example Let s find the inverse of the following relation:
f (x) = x + 2 If we call the dependent variable y, then we can rewrite our relation as y=x+2 Swapping the names of the variables, we get x=y+2 which can be manipulated with algebra to obtain y=x 2 If we replace the new variable y by the relation notation f 1(x), we get f 1(x) = x 2 The domain and range of the original relation f both span the entire set of real numbers Therefore, the domain and range of the inverse relation f 1 also both span the entire set of reals
Another example Let s find the inverse of the following relation:
g(x) = (x1/2) If we call the dependent variable y, then we can rewrite the relation as y = (x1/2) When we switch the names of the variables, we get x = ( y1/2) Squaring both sides produces x2 = y Reversing the left- and right-hand sides gives us y = x2