# Introduction to the History of Mathematics by Howard Eves

By Howard Eves

This vintage best-seller through a well known writer introduces arithmetic background to math and math schooling majors. recommended essay themes and challenge reviews problem scholars. CULTURAL CONNECTIONS sections clarify the time and tradition within which arithmetic built and advanced. pictures of mathematicians and fabric on girls in arithmetic are of targeted curiosity.

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Example text

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.