In Problems 25–34 find the half-range cosine and sine expansions of the given function.
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Differential Equations with Boundary-Value Problems (MindTap Course List)
- In Problems 27–36, verify that the functions f and g are inverses of each other by showing that f(g(x)) = x and g(f(x)) any values of x that need to be excluded. = x. Give 27. f(x) = 3x + 4; g(x) = (x- 4) 28. f(x) = 3 – 2x; g(x) = -(x – 3) 29. f(x) = 4x – 8; 8(x) = + 2 30. f(x) = 2x + 6; 8(x) = ;x - 3 31. f(x) = x' - 8; g(x)· Vx + 8 32. f(x) = (x – 2)², 2; g(x) = Vĩ + 2 33. f(x) = ; 8(x) = 34. f(x) = x; g(x) x - 5 2x + 3' 2x + 3 4x - 3 3x + 5 35. f(x) *: 8(x) = 8(x) 36. f(x) = 1- 2x x + 4 2 - x 1.7 82 CHAPTER 1 Graphs and Functions In Problems 37-42, the graph of a one-to-one function f is given. Draw the graph of the inverse function f"1. For convenience (and as a hint), the graph of y = x is also given. 37. y= X 38. 39. y =X 3 (1, 2), (0, 1) (-1,0) (2. ) (2, 1) (1, 0) 3 X (0, -1) -3 (-1, -1) 3 X -3 (-2, -2) (-2, -2) -하 -하 -하 40. 41. y = x 42. y = X (-2, 1). -3 3 X (1, -1)arrow_forwardIn Problems 11–20, for the given functions f and g. find: (a) (f° g)(4) (b) (g•f)(2) (c) (fof)(1) (d) (g ° g)(0) \ 11. f(x) = 2x; g(x) = 3x² + 1 12. f(x) = 3x + 2; g(x) = 2x² – 1 1 13. f(x) = 4x² – 3; g(x) = 3 14. f(x) = 2x²; g(x) = 1 – 3x² 15. f(x) = Vx; 8(x) = 2x 16. f(x) = Vx + 1; g(x) = 3x %3D 1. 17. f(x) = |x|; g(x) = 18. f(x) = |x – 2|: g(x) x² + 2 2 x + 1 x² + 1 19. f(x) = 3 8(x) = Vĩ 20. f(x) = x³/2; g(x) = X + 1'arrow_forwardIn Problems 13–22, for the given functions f and g, find:(a) (f ∘ g) (4) (b) (g ∘ f) (2) (c) (f ∘ f) (1) (d) (g ∘ g) (0)arrow_forward
- Problem 13 (#2.3.10).Determine whether each of these functions from {a,b,c,d} to itself are one-to-one. a) f(a) = b, f(b) = a, f (c) = c, f(d) = d, b) f(a) = b, f(b) = b, f(c) = d, f(d) = c, c) f(a) = d, f(b) = b, f (c) = c, f (d) = d.arrow_forward.4 Evaluate J2 sec -1 Va drarrow_forwardProblem 4. Let f and g be functions on [a, b], and assume that f(a) = 1 = g(b) and f(b) = 0 = g(a). Show that {f,g} is independent in F[a, b].arrow_forward
- In Problems 33–44, determine algebraically whether each function is even, odd, or neither. 34. f(x) = 2x* –x? 38. G(x) = Vĩ 33. f(x) = 4x 37. F(x) = V 35. g(x) = -3x² – 5 39. f(x) = x + |x| 36. h (х) — Зx3 + 5 40. f(x) = V2r²+ 1 x² + 3 -x 42. h(x) =- 1 2x 44. F(x) 41. g(x) 43. h(x) x2 - 1 3x2 - 9arrow_forwardQuestion 10. Plot the result of the following convolution X₁(t) 0 2 4 * X2(t) 0 2 6arrow_forward* 1.3 • If f(x) = 1. D; = R Rf = {1,0} 2. D; = [1,00) Rf = [0, c0) 3. D; = R R; = {1,–1} 4. D; = [1,00) R; = {1,0} then, %3D %3D %3Darrow_forward
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