The Theory of Matrices [Vol 2] by Felix R. Gantmacher

By Felix R. Gantmacher

This treatise, through one among Russia's top mathematicians, provides in simply available shape a coherent account of matrix conception for you to purposes in arithmetic, theoretical physics, information, electric engineering, and so on. the person chapters were saved so far as attainable self sustaining of one another, in order that the reader conversant in the contents of bankruptcy 1 can continue instantly to the chapters of distinctive curiosity. a lot of the fabric has been to be had beforehand merely within the periodical literature.

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Y ϭ f ͑x͒ Ϫ 5 41. y ϭ Ϫf ͑Ϫx͒ Ϫ 2 42. y ϭ Ϫf ͑x Ϫ 4͒ 43. y ϭ f ͑x ϩ 6͒ ϩ 2 44. y ϭ f ͑x Ϫ 1͒ ϩ 3 45. Use the graph of f ͑x͒ ϭ Ίx to sketch the graph of each function. In each case, describe the transformation. (a) y ϭ Ίx ϩ 2 (b) y ϭ Ϫ Ίx ΂ ␲ ϩ1 2 ΃ (c) y ϭ Ίx Ϫ 2 (b) h͑x͒ ϭ Ϫsin͑x Ϫ 1͒ 47. Given f ͑x͒ ϭ Ίx and g͑x͒ ϭ x 2 Ϫ 1, evaluate each expression. (a) f ͑g͑1͒͒ (b) g͑ f ͑1͒͒ (c) g͑ f ͑0͒͒ (d) f ͑g͑Ϫ4͒͒ (e) f ͑g͑x͒͒ (f) g͑ f ͑x͒͒ ΂ ΂␲4 ΃΃ (d) g f ΂ ΂12΃΃ 2 −2 g x 2 −2 4 (b) f g (c) g͑ f ͑0͒͒ (e) f ͑g͑x͒͒ (f) g͑ f ͑x͒͒ 54.

Y 41. 60͑50 Ϫ x͒, where x is the number of pounds of the less expensive commodity. (b) Find the inverse function of the cost function. What does each variable represent in the inverse function? (c) Use the context of the problem to determine the domain of the inverse function. (d) Determine the number of pounds of the less expensive commodity purchased if the total cost is $73. C ϭ 9 ͑F Ϫ 32͒, 44. 6, represents the Celsius temperature C as a function of Fahrenheit temperature F. 5 (a) Find the inverse function of C.

Use the definition of an inverse function to check the following. f ͑x͒ ϭ x ϩ c f ͑x͒ ϭ cx and and f Ϫ1͑x͒ ϭ x Ϫ c are inverse functions of each other. x f Ϫ1͑x͒ ϭ , c 0, are inverse functions of each other. 3 EXAMPLE 1 Inverse Functions 21 Verifying Inverse Functions Show that the functions are inverse functions of each other. f ͑x͒ ϭ 2x 3 Ϫ 1 g͑x͒ ϭ and Ίx ϩ2 1 3 Solution Because the domains and ranges of both f and g consist of all real numbers, you can conclude that both composite functions exist for all x.

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