ISM_T11_C07_A - CHAPTER 7 TRANSCENDENTAL FUNCTIONS 7.1...

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CHAPTER 7 TRANSCENDENTAL FUNCTIONS 7.1 INVERSE FUNCTIONS AND THEIR DERIVATIVES 1. Yes one-to-one, the graph passes the horizontal test. 2. Not one-to-one, the graph fails the horizontal test. 3. Not one-to-one since (for example) the horizontal line y intersects the graph twice. œ# 4. Not one-to-one, the graph fails the horizontal test. 5. Yes one-to-one, the graph passes the horizontal test 6. Yes one-to-one, the graph passes the horizontal test 7. Domain: 0 x 1, Range: 0 y 8. Domain: x 1, Range: y 0 ±Ÿ Ÿ ± ² 9. Domain: 1 x 1, Range: y 10. Domain: x , Range: y ³ŸŸ ³ _ ±±_ ³±Ÿ 11 ## 11. The graph is symmetric about y x. œ (b ) y 1 x y 1 x 1 y x 1 y 1 x f (x ) œ ³Êœ ³Ê œ œ ³ œ ÈÈ È # ± "
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426 Chapter 7 Transcendental Functions 12. The graph is symmetric about y x. œ y x y f ( x ) œÊœÊœœ """ ±" xyx 13. Step 1: y x 1 x y 1 x y 1 œ±Ê œ²Êœ ² ## È Step 2: y x 1 f (x) œ² œ È ±" 14. Step 1: y x y, since x . œ Ê œ² Ÿ! # È Step 2: y x f (x) œ È ±" 15. Step 1: y x 1 x y 1 x (y 1) œ²Ê œ±Êœ± $ $ "Î$ Step 2: y x 1 f (x) œ± œ $ ±" È 16. Step 1: y x 2x 1 y (x 1) y x 1, since x 1 x 1 y œ²±Êœ² Ê œ²  Êœ± ÈÈ S t e p 2 : y1 xf( x ) œ± œ È ±" 17. Step 1: y (x y x 1, since x 1 x y 1 œ± Ê œ±   ²Êœ ² # Step 2: y x 1 f (x) œ È ±" 18. Step 1: y x x y œÊ œ #Î$ $Î# Step 2: y x f (x) œœ $Î# ±" 19. Step 1: y x y œ & "Î& Step 2: y x f (x); & ±" È Domain and Range of f : all reals; ±" f f (x) x x and f (f(x)) x x ab a b ˆ‰ ±" "Î& ±" & & "Î& 20. Step 1: y x y œ % "Î% Step 2: y x f (x); % ±" È Domain of f : x 0, Range of f : y 0; ±" ±"    f f (x) x x and f (f(x)) x x a b ±" "Î% ±" % % "Î% 21. Step 1: y x 1 x y 1 x (y 1) œ±Ê œ²Êœ² $ $ "Î$ Step 2: y x 1 f (x); œ $ ±" È Domain and Range of f : all reals; ±" f f (x) (x 1) 1 (x 1) 1 x and f (f(x)) x 1 1 x x a ba b ±" "Î$ ±" $ $ $ "Î$ "Î$ ± œ ² ± œ œ ± ² œ œ
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Section 7.1 Inverse Functions and Their Derivatives 427 22. Step 1: y x x y x 2y 7 œ± Ê œ ² Ê œ ² "" ## # # 77 Step 2: y 2x 7 f (x); œ²œ ±" Domain and Range of f : all reals; ±" f f (x) (2x 7) x x and f (f(x)) 2 x 7 (x 7) 7 x ab ˆ‰ ˆ ±" ±" # # # # œ ²±œ²±œ œ ±²œ±²œ 7 7 23. Step 1: y x x œÊœ Ê œ """ # xyy # È Step 2: y f (x) œœ " ±" È x Domain of f : x 0, Range of f : y 0; ±" ±" ³³ f f (x) x and f (f(x)) x since x 0 ±" ±" " " œ œ œ œ ³ Š‹ É " È x xx # " # x 24. Step 1: y x x Ê œ $ xy y $ "Î$ Step 2: y f (x); œ $ ±" x x "Î$ É Domain of f : x 0, Range of f : y 0; ±" ±" ÁÁ f f (x) x and f (f(x)) x ±" ±" " " ±"Î$ ±" œ œ x x "Î$ $ " $ 25. (a) y 2x 3 2x y 3 œ²Ê œ± x f (x) Êœ±Ê y 3x 3 ±" (c) 2, ¸ ¹ df df dx dx x1 oe oe " " # (b) 26. (a) y x 7 x y 7 œ² Ê œ ± 55 x 5y 35 f (x) 5x 35 Êœ ± Ê œ ± ±" (c) , 5 ¸ ¹ df df dx 5 dx x oe oe$%Î& " " (b) 27. (a) y 5 4x 4x 5 y œ± Ê œ± x f (x) x 44 y ±" (c) 4, ¸ ¹ df df dx dx 4 x3 oeÎ# oe " " (b)
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428 Chapter 7 Transcendental Functions 28. (a) y 2x x y œÊ œ ## " # x y f (x) Êœ Ê œ " ±" # È 2 x È È (c) 4x 20, ¸ k df dx x x5 oe& oe œœ x ¹¹ df dx 0 2 " x0 0 oe& oe "" # ±"Î# # È (b) 29. (a) f(g(x)) x x, g(f(x)) x x œ œ ˆ‰ È È $ $ $ $ (c) f (x) 3x f (1) 3, f ( 1) 3; w# w w œ± œ g (x) x g (1) , g ( 1) w ±#Î$ w w " œ ± œ 33 3 (d) The line y 0 is tangent to f(x) x at ( ); œ œ !ß!
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ISM_T11_C07_A - CHAPTER 7 TRANSCENDENTAL FUNCTIONS 7.1...

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