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Illustration for Question and Answer 11-7
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variable is represented by r (the radial distance from the origin) How can we tell if the relation is a function of q
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Answer 11-8
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We can draw the graph of the relation in a Cartesian plane, plotting values of q along the horizontal axis, and plotting values of r along the vertical axis We can allow both q and r to attain all possible real-number values Then we can use the Cartesian vertical-line test to see if the relation is a function of q
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Question 11-9
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To add one function to another, we add both sides of their equations This can be done in either order, producing identical results If f1 and f2 are functions of x, then ( f1 + f2)(x) = f1(x) + f2(x) and ( f2 + f1)(x) = f2(x) + f1(x)
Review Questions and Answers
To subtract one function from another, we subtract both sides of their equations This can be done in either order, usually producing different results If f1 and f2 are functions of x, then ( f1 f2)(x) = f1(x) f2(x) and ( f2 f1)(x) = f2(x) f1(x) To multiply one function by another, we multiply both sides of their equations This can be done in either order, producing identical results If f1 and f2 are functions of x, then ( f1 f2)(x) = f1(x) f2(x) and ( f2 f1)(x) = f2(x) f1(x) To divide one function by another, we divide both sides of their equations This can be done in either order, usually producing different results If f1 and f2 are functions of x, then ( f1/f2)(x) = f1(x)/f2(x) and ( f2/f1)(x) = f2(x)/f1(x)
Question 11-10
When we add, subtract, multiply, or divide functions, we must adhere to three important rules What are they
Answer 11-10
First, we must be sure that the functions both operate on the same thing In other words, the independent variables must describe the same parameters or phenomena Second, we must restrict the domain of the resultant function to only those values that are in the domains of both functions (the intersection of the domains) Third, if we divide a function by another function, we can t define the resultant function for any value of the independent variable where the denominator function becomes 0
12
Question 12-1
How can we informally define the inverse of a relation
Part Two 403 Answer 12-1
The inverse of a relation is another relation that undoes whatever the original relation does Also, the original relation undoes whatever its inverse does
Question 12-2
How can we rigorously define the inverse of a relation
Answer 12-2
Let f be a relation where x is the independent variable and y is the dependent variable The inverse relation for f is another relation f 1 such that f 1 [ f (x)] = x for all values of x in the domain of f, and f [f for all values of y in the range of f
Question 12-3
(y)] = y
Suppose we ve drawn the graph of a relation f in the Cartesian xy plane How can we create the graph of the inverse relation f 1
Answer 12-3
Imagine the line y = x as a point reflector For any point on the graph of f, its counterpoint on the graph of f 1 lies on the opposite side of the line y = x but the same distance away, as shown in Fig 20-2 Mathematically, we can do this transformation by reversing the sequence of the ordered pair representing the point When we want to obtain the graph of f 1 based on the graph of f, we can flip the whole graph over in three dimensions around the line y = x, as if that line were the hinge of a revolving door
Question 12-4
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