# Notes on Rubik's Magic Cube by David Singmaster

By David Singmaster

Notes on Rubik's 'Magic dice'

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Notes on Rubik's Magic Cube

Notes on Rubik's 'Magic dice'

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But the question arises: Which curves in the xy-plane are graphs of functions? This is answered by the following test. The Vertical Line Test A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. The reason for the truth of the Vertical Line Test can be seen in Figure 13. If each vertical line x ෇ a intersects a curve only once, at ͑a, b͒, then exactly one functional value is defined by f ͑a͒ ෇ b. But if a line x ෇ a intersects the curve twice, at ͑a, b͒ and ͑a, c͒, then the curve can’t represent a function because a function can’t assign two different values to a.

87 The function P is typical of the functions that arise whenever we attempt to apply calculus to the real world. We start with a verbal description of a function. Then we may be able to construct a table of values of the function, perhaps from instrument readings in a scientific experiment. Even though we don’t have complete knowledge of the values of the function, we will see throughout the book that it is still possible to perform the operations of calculus on such a function. C. Again the function is described in words: C͑w͒ is the cost of mailing a large envelope with weight w.

A) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of \$14,000? On \$26,000? (c) Sketch the graph of the total assessed tax T as a function of the income I. com 67–72 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. x2 x4 ϩ 1 67. f ͑x͒ ෇ x x2 ϩ 1 68. f ͑x͒ ෇ 69. f ͑x͒ ෇ x xϩ1 70. f ͑x͒ ෇ x x 71. f ͑x͒ ෇ 1 ϩ 3x 2 Ϫ x 4 Խ Խ 72. f ͑x͒ ෇ 1 ϩ 3x 3 Ϫ x 5 73. If f and t are both even functions, is f ϩ t even?