Download The Representation Theory of the Symmetric Groups by G. D. James (auth.) PDF
By G. D. James (auth.)
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Additional info for The Representation Theory of the Symmetric Groups
Sample text
This function is defined only for x-values for which 4 Ϫ 3x Ն 0. By solving this inequality, you can conclude that x Յ 43. So, the domain is the interval ͑Ϫ ϱ, 43͔. Now try Exercise 73. In Example 7(c), note that the domain of a function may be implied by the physical context. For instance, from the equation 4 V ϭ 3 r 3 you would have no reason to restrict r to positive values, but the physical context implies that a sphere cannot have a negative or zero radius. 4 Functions 45 Applications h r =4 r Example 8 The Dimensions of a Container You work in the marketing department of a soft-drink company and are experimenting with a new can for iced tea that is slightly narrower and taller than a standard can.
5, Ϫ6͒, m ϭ 1 ͑Ϫ8, 1͒, m is undefined. ͑1, 5͒, m is undefined. 48. ͑Ϫ5, 4͒, m ϭ 2 1 50. ͑7, Ϫ2͒, m ϭ 2 ͑3, Ϫ2͒, m ϭ 0 ͑10, Ϫ6͒, m ϭ Ϫ1 ͑0, Ϫ9͒, m ϭ Ϫ2 ͑Ϫ1, Ϫ6͒, m ϭ Ϫ 12 In Exercises 51– 64, find the slope-intercept form of the equation of the line that passes through the given point and has the indicated slope m. Sketch the line. 51. ͑0, Ϫ2͒, m ϭ 3 53. ͑Ϫ3, 6͒, m ϭ Ϫ2 55. ͑4, 0͒, m ϭ Ϫ 13 57. 59. 60. 61. 63. 52. ͑0, 10͒, m ϭ Ϫ1 54. ͑0, 0͒, m ϭ 4 56. ͑8, 2͒, m ϭ 14 58. ͑Ϫ2, Ϫ5͒, m ϭ 34 ͑2, Ϫ3͒, m ϭ Ϫ 12 ͑6, Ϫ1͒, m is undefined.
Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is, m1 ϭ Ϫ1͞m2. Example 4 y 2x − 3y = 5 3 2 Finding Parallel and Perpendicular Lines Find the slope-intercept forms of the equations of the lines that pass through the point ͑2, Ϫ1͒ and are (a) parallel to and (b) perpendicular to the line 2x Ϫ 3y ϭ 5. 40 2x Ϫ 3y ϭ 5 Write original equation. Ϫ3y ϭ Ϫ2x ϩ 5 yϭ 2 3x Ϫ Subtract 2x from each side. 5 3 Write in slope-intercept form. 40. a.