ISM_T11_C01_I - 56 Chapter 1 Preliminaries 41. (a) The...

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56 Chapter 1 Preliminaries 41. (a) The function is defined for all values of x, so the domain is . ab ±_ß _ (b) Since x attains all nonnegative values, the range is . l l Ò±#ß _Ñ 42. (a) Since the square root requires x , the domain is . " ±   ! б_ß "Ó # (b) Since x attains all nonnegative values, the range is . È " ± Ò±#ß _Ñ 43. (a) Since the square root requires x , the domain is . "' ±   ! Ò±%ß %Ó # (b) For values of x in the domain, x , so x . The range is . ! Ÿ "' ± Ÿ "' ! Ÿ "' ± Ÿ % Ò!ß %Ó # # È 44. (a) The function is defined for all values of x, so the domain is . ±_ß _ (b) Since attains all positive values, the range is . $" ß _ x 45. (a) The function is defined for all values of x, so the domain is . ±_ß _ (b) Since e attains all positive values, the range is . # ±$ß _ ± x 46. (a) The function is equivalent to y tan x, so we require x for odd integers k. The domain is given by x for œ# # Á Á kk 11 #% odd integers k. (b) Since the tangent function attains all values, the range is . ±_ß _ 47. (a) The function is defined for all values of x, so the domain is . ±_ß _ (b) The sine function attains values from to , so sin x and hence sin x . The ±" " ±#Ÿ# $ ² Ÿ# ±$Ÿ# ±"Ÿ" range is 3 1 . Ò± ß Ó 48. (a) The function is defined for all values of x, so the domain is . ±_ß _ (b) The function is equivalent to y x , which attains all nonnegative values. The range is . œ Ò!ß _Ñ È & # 49. (a) The logarithm requires x , so the domain is . ± $ ³ ! $ß _ (b) The logarithm attains all real values, so the range is . ±_ß _ 50. (a) The function is defined for all values of x, so the domain is . ±_ß _ (b) The cube root attains all real values, so the range is . ±_ß _ 51. (a) The function is defined for x , so the domain is . ±% Ÿ Ÿ % Ò±%ß %Ó (b) The function is equivalent to y x , x , which attains values from to for x in the domain. The œl l ± % ŸŸ % !# È range is . Ò!ß #Ó 52. (a) The function is defined for x , so the domain is . ±# Ÿ Ÿ # Ò±#ß #Ó (b) The range is . Ò±"ß "Ó 53. First piece: Line through and . m y x x ! ß" " ß! œ œ œ±"Ê œ± ²"œ"± !±" ±" "±! " Second piece: Line through and . m y x x x a b " # œ œ œ±"Ê œ± ±" ²"œ± ²#œ#± #±" " fx x, x x, x œ œ !Ÿ ´" "Ÿ Ÿ# 54. First piece: Line through and 2 5 . m y x !ß ! ß œ œ Ê œ 55 5 22 2 ±! Second piece: Line through 2 5 and 4 . m y x 2 5 x 10 10 a b ß œ œ œ± Ê œ± ± ² œ± ² œ ± ± ± 5 5 5 5 x 42 2 2 2 2 2 f x (Note: x 2 can be included on either piece.) x, x 2 1 0 ,2x4 ± œœ !Ÿ ´ ±Ÿ Ÿ 5 2 5x 2
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Chapter 1 Practice Exercises 57 55. (a) f g f g f f ab ab a b a b Š‹ ‰ ± ± " œ œ "œ œ" "" ±"²# " È (b) g f g f g or a b a b ˆ‰ É ‰#œ #œ œ œ " # ²# #Þ& & 2 É È " # (c) f f x f f x f x, x a ‰œ œœ œ Á ! xx (d) g g x g g x g a œ œ œ ²# "²# È ÉÉ È È x x x " # % È x 56. (a) f g f g f f b a b È ± ± "²"œ !œ#±!œ# $ (b) g f f g g g a b a b a b È ‰#œ #œ# ± #œ!œ! ² " œ " $ (c) f f x f f x f x x x a b a b a b œ œ #± œ#± #± œ (d) g g x g g x g x x a ÈÈ É œ œ ²" œ ²"²" $$ $ 57. (a) f g x f g x f x x x, x . a ˆ‰ˆ‰ œ œ ²# œ#± ²# œ±  ±# # g f x f g xgx x x a b a b a b È È œ # ± œ # ± ² # œ % ± # ## (b) Domain of f g: ‰ Ò±#ß _ÑÞ Domain of g f: ‰ Ò±#ß #ÓÞ
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ISM_T11_C01_I - 56 Chapter 1 Preliminaries 41. (a) The...

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