By Ron Larson
Crucial Calculus: Early Transcendental features responds to the becoming call for for a extra streamlined and swifter paced textual content at a lower cost for college kids. this article maintains the Larson culture via providing teachers confirmed pedagogical strategies and obtainable content material and cutting edge studying assets for pupil good fortune.
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Extra resources for Essential Calculus: Early Transcendental Functions
Arccos 0 85. arccos 87. arctan Ί3 3 89. arccsc͑Ϫ Ί2 ͒ 4 3 2 1 x −3 −2 −1 ,π 4 In Exercises 83–90, evaluate the expression without using a calculator. y 4 3 2 1 ( ) 1 2 3 4 In Exercises 77 and 78, use the graph of the function f to sketch the graph of f ؊1. com. 77. ( x −3 −2 4 y , −π 6 29 y = arctan x (− x 1 π 2 y 76. 4 3 2 1 −3 −2 −1 82. 8 1 88. arccot͑Ϫ Ί3 ͒ 90. arccos Ϫ Ί3 2 In Exercises 91–94, use a calculator to approximate the value. Round your answer to two decimal places. 91. 8͒ 92.
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. The composite of f with g is given by f ͑ g ͑x͒͒ ϭ 2 ϭ2 y 3 x+1 2 Ί͑2x 2x ϭΊ 2 g͑ f ͑x͒͒ ϭ x 1 −2 x ϩ2 1 Ϫ 1 The composite of g with f is given by 1 −2 3 3 ϭxϩ1Ϫ1 ϭ x. y=x 2 g(x) = Ίx ϩ2 1 Ϫ 1 2 3 3 Ϫ 1͒ ϩ 1 2 3 3 f(x) = 2x 3 − 1 f and g are inverse functions of each other. 22 3 x3 ϭ Ί ϭ x. 22). STUDY TIP In Example 1, try comparing the functions f and g verbally.
F ͑x͒ ϭ arcsin ͑x Ϫ 1͒ 132. f ͑x͒ ϭ arctan x ϩ 133. f ͑x͒ ϭ arcsec 2x 134. f ͑x͒ ϭ arccos (b) cos͑arcsecΊ5͒ 53΅ 5 (b) tan΄ arcsinϪ ΅ 6 ΄ 110. (a) sec arctan Ϫ In Exercises 111–114, solve the equation for x. 1 111. arcsin͑3x Ϫ ͒ ϭ 2 112. arctan͑2x Ϫ 5͒ ϭ Ϫ1 113. arcsin Ί2x ϭ arccos Ίx 114. arccos x ϭ arcsec x 136. Prove that if f has an inverse function, then ͑ f Ϫ1͒Ϫ1 ϭ f. 137. Prove that if a function has an inverse function, then the inverse function is unique. 138. Prove that a function has an inverse function if and only if it is one-to-one.