Concept explainers
In Problems 37–40 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x).
37.
Want to see the full answer?
Check out a sample textbook solutionChapter 2 Solutions
A First Course in Differential Equations with Modeling Applications (MindTap Course List)
- In 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 2–4, for the given functions fand g find: (a) (f° g) (2) (b) (g • f)(-2) (c) (fo f) (4) (d) (g ° 8) (-1) 2. f(x) = 3x – 5; g(x) = 1 – 2r 3. f(x) = Vx + 2: g(x) = 2x² + 1 4. f(x) = e"; g(x) = 3x – 2arrow_forwardIn Problems 23–28, answer the questions about the given function. x² + 2 26. f(x) = x + 4 23. f(x) = 2x? - x - 1 (a) Is the point (-1, 2) on the graph of f? (b) If x = -2, what is f(x)? What point is on the graph of f? (c) If f(x) = -1, what is x? What point(s) are on the graph of f? (d) What is the domain of f? (e) List the x-intercepts, if any, of the graph of f. (f) List the y-intercept, if there is one, of the graph of f. 24. f(x) = -3x² + 5x (a) Is the point (-1, 2) on the graph of f? (b) If x = -2, what is f(x)? What point is on the graph of f? (c) If f(x) = -2, what is x? What point(s) are on the graph of f? (d) What is the domain of f? (e) List the x-intercepts, if any, of the graph of f. (f) List the y-intercept, if there is one, of the graph of f. x + 2 (a) Is the point ( 1,) on the graph of f? (b) If x = 0, what is f(x)? What point is on the graph of f? (c) If f(x) =5. what is x? What point(s) are on the graph of f? (d) What is the domain of f? (e) List the x-intercepts, if…arrow_forward
- In Problems 19–30, graph the function f by starting with the graph of y = x² and using transformations (shifting, compressing, stretching, and/or reflection). Verify your results using a graphing utility. [Hint: If necessary, write f in the form f(x) = a(x – h)² + k.] 19. f(x) = 20. f(x) = 2x2 + 4 21. f(x) = (x + 2)² – 2 22. f(x) = (x – 3)² – 10 23. f(x) = x² + 4x + 2 24. f(х) — х? — бх — 1 25. f(x) = 2x? – 4x + 1 26. f(x) = 3x? + 6x 4 27. f(x) = -x² - 2x 28. f(x) 3D-2х? + 6х + 2 29, f(x) : 30. f(x) 1 + xarrow_forwardIn Problems 29–40: (a) Find the domain of each function. (d) Based on the graph, find the range. (b) Locate any intercepts. (e) Is f continuous on its domain? S3x 14 (c) Graph each function. S2r 29. f(x) : if x + 0 S-2x + 3 3x – 2 if x 1 x + 3 2x + 5 if -3 sx0 S1 + x if x 0 35. f(x) : 36. f(x) = 37. f(x) if x 20 S2 - x if -3 sx1arrow_forwardIn 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_forward
- In Problems 23–30, use the given zero to find the remaining zeros of each function.arrow_forwardIn Problems 43–66, find the indicated extremum of each function on the given interval.arrow_forward23. What is the domain of the function f(x) = Vx² – 16? %3D In Problems 25–32, use the given functions f and g. (a) Solve f(x) = 0. (e) Solve g(x) s 0. (b) Solve g(x) = 0. (f) Solve f(x) >g(x).arrow_forward
- In Problems 19–22, construct a polynomial function f that has the given properties. There is no unique answer. 19. f is of degree 4, its graph is symmetric with respect to the y-axis, y-intercept is (0, –6)arrow_forwardIn Problems 31–42:(a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function.(d) Based on the graph, find the range. (e) Is f continuous on its domain?arrow_forwardIn 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_forward
- Discrete Mathematics and Its Applications ( 8th I...MathISBN:9781259676512Author:Kenneth H RosenPublisher:McGraw-Hill EducationMathematics for Elementary Teachers with Activiti...MathISBN:9780134392790Author:Beckmann, SybillaPublisher:PEARSON
- Thinking Mathematically (7th Edition)MathISBN:9780134683713Author:Robert F. BlitzerPublisher:PEARSONDiscrete Mathematics With ApplicationsMathISBN:9781337694193Author:EPP, Susanna S.Publisher:Cengage Learning,Pathways To Math Literacy (looseleaf)MathISBN:9781259985607Author:David Sobecki Professor, Brian A. MercerPublisher:McGraw-Hill Education