advertisement

Pre-Calculus Chapter 2 Test Review Name________________________ Assignment: Use the function 𝒇(𝒙) = 𝟑𝒙𝟐 − 𝟏 to evaluate the following. 2. f(x – 3) 1. f(-3) 3. f(a + h) 4. 𝑓(𝑎+ℎ)−𝑓(𝑎) ℎ Find the domain of the function. Write your answer in interval notation. 5. 𝑓(𝑥) = 3 𝑥 2 +𝑥−6 6. 𝑓(𝑥) = √𝑥 + 9 7. 𝑓(𝑥) = 5 √𝑥−2 8. 𝑓(𝑥) = 𝑥 6 + 𝑥−1 𝑥−7 9. 𝑓(𝑥) = √𝑥 2 − 4 10. 𝑓(𝑥) = 𝑥 2 + 2𝑥 − 3 𝒊𝒇 𝒙 < 0 𝟑𝒙 Given: 𝒇(𝒙) = { 𝒙 + 𝟏 𝒊𝒇 𝟎 ≤ 𝒙 ≤ 𝟐. Evaluate the following: (𝒙 − 𝟐)𝟐 𝒊𝒇 𝒙 > 2 11. f(-3) 12. f(0) 13. f(2) 14. f(4) Sketch the graph of the piece-wise function. 𝑥 + 2 𝑖𝑓 𝑥 < −2 15. 𝑓(𝑥) = { 2 𝑖𝑓 𝑥 ≥ −2 𝑥 𝑖𝑓 𝑥 < −2 4 2 𝑖𝑓 − 2 ≤ 𝑥 < 1 16. 𝑓(𝑥) = { 𝑥 𝑖𝑓 𝑥 ≥ 1 −2𝑥 + 1 𝑖𝑓 𝑥 ≤ 0 𝑥+4 2 17. 𝑓(𝑥) = { 4 − 𝑥 𝑖𝑓 0 < 𝑥 ≤ 2 𝑖𝑓 𝑥 > 2 −𝑥 + 4 Describe the transformations that would be applied to the graph of 𝒇(𝒙) = |𝒙|. 1 18. 𝑓(𝑥) = 5 |𝑥 + 3| − 7 1 19. 𝑓(𝑥) = 6|−3𝑥| 20. 𝑓(𝑥) = − |2 𝑥| + 1 A function f is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. 21. 𝑓(𝑥) = 𝑥 3 ; shift 4 units to the left and shift downward 1 unit. 22. 𝑓(𝑥) = |𝑥|; reflect over the x-axis, shift to the right 1 unit, stretch vertically by a factor of 3. 1 23. 𝑓(𝑥) = 𝑥 2; stretch horizontally by a factor of 2, shrink vertically by a factor of 4, shift up 3. 1 24. 𝑓(𝑥) = √𝑥; reflect over the y-axis, shrink horizontally by a factor of 3. The graph of f is given. Use it to graph each of the following functions. 25. 𝑦 = −𝑓(2𝑥) + 1 26. 𝑦 = 2𝑓(𝑥 + 1) 1 2 27. 𝑦 = 𝑓 (− 𝑥) − 3 Convert the quadratic to standard form, then find the following: 28. 𝑓(𝑥) = 2𝑥 2 + 12𝑥 + 12 29. 𝑓(𝑥) = −𝑥 2 − 3𝑥 + 3 Vertex: Vertex: Max/Min Value: Max/Min Value: Domain: Domain: Range: Range: x-intercept: x-intercept: y-intercept: y-intercept: 30. Find the minimum or maximum value of 𝑓(𝑥) = 10𝑥 2 + 40𝑥 + 113. 31. A soft-drink vendor at a popular beach analyzes his sales records, and finds that if he sells x cans of soda in one day, his profit (in dollars) is given by 𝑃(𝑥) = −0.001𝑥 2 + 3𝑥 − 1800. What is his maximum profit per day, and how many cans of soda must he sell to reach that profit? 32. A poster is 8 inches longer than it is wide. Find a function that models the area in terms of the length of one of its sides. 33. The sum of two positive numbers is 50. Find a function that models their product in terms of one of the numbers. 34. If the perimeter of a rectangular garden is to be 480 ft., find the dimensions of the garden that would give the largest area. What is the maximum area? 35. A rancher wants to build a rectangular pig pen with an area of 100 m2. a) Find a function that models the length of fencing required. b) Find the pen dimensions that would require the minimum amount of fencing. 36. A farmer only has 1800 ft. of fencing available to build five adjacent pens as shown in the diagram below. a) Express the total area of the pens as a function of x. x y b) What value of x will maximize the total area? Given 𝒇(𝒙) = 𝟑𝒙 − 𝟓 and 𝒈(𝒙) = 𝟐 − 𝒙𝟐 , find the following: 37. f + g 38. f – g 39. fg 40. 𝑓 ∘ 𝑔 41. 𝑔 ∘ 𝑓 42. 𝑓 ∘ 𝑓 43. (𝑓 ∘ 𝑔)(−4) 44. (𝑔 ∘ 𝑔)(−2) Express the function in the form 𝒇 ∘ 𝒈. 1 45. 𝐹(𝑥) = (𝑥 + 3)6 46. 𝐺(𝑥) = 𝑥−7 47. 𝐻(𝑥) = √1 − 𝑥 Determine whether the function is one to one. 48. 𝑓(𝑥) = 3𝑥 − 1 1 49. 𝑓(𝑥) = 2𝑥 − 𝑥 2 50. 𝑓(𝑥) = 𝑥 52. 𝑓(𝑥) = (𝑥 + 1)3 53. 𝑓(𝑥) = √𝑥 − 2 Find the inverse of the function. 51. 𝑓(𝑥) = 2𝑥+1 3 Use the Inverse Function Property to determine whether f and g are inverses of each other. 54. 𝑓(𝑥) = 3𝑥 and 𝑔(𝑥) = 𝑥 3 55. 𝑓(𝑥) = 1−𝑥 4 and 𝑔(𝑥) = 1 − 4𝑥 Answers to Chapter 2 Review Pre-Calculus 1. 26 2. 3𝑥 2 − 18𝑥 + 26 3. 3𝑎2 + 6𝑎ℎ + 3ℎ2 − 1 6. [−9, ∞) 7. (2, ∞) 8. (−∞, 1) ∪ (1,7) ∪ (7, ∞) 11. -9 12. 1 13. 3 17. 5. (−∞, −3) ∪ (−3,2) ∪ (2, ∞) 4. 6a + 3h 14. 4 9. (−∞, −2] ∪ [2, ∞) 15. 10. (−∞, ∞) 16. 18. Shift left 3, vertical shrink 1 by 5, shift down 7 1 3 19. Reflect over the y-axis, horizontal shrink by , vertical stretch by 6 20. Horizontal stretch by 2, reflect over the x-axis, shift up 1 1 1 4 2 22. 𝑓(𝑥) = −3|𝑥 − 1| 25. 26. 3 2 29. 𝑓(𝑥) = − (𝑥 + 2) + 30. Min: 73 21 4 3 21 ); 4 ; V: (− 2 , Max: 21 ; 4 D: (−∞, ∞); R: (−∞, 31. 1500 cans, $450 max profit 36a) 𝐴(𝑥) = −3𝑥 2 + 900𝑥 36b) 150ft 41. −9𝑥 2 + 30𝑥 − 23 45. f(x) = 𝑥 6 ; g(x) = x + 3 50. yes 24. 𝑓(𝑥) = √−3𝑥 28. 𝑓(𝑥) = 2(𝑥 + 3)2 − 6 ; V: (-3, -6); Min: -6 D: (−∞, ∞); R: [−6, ∞); x-int: (−3 ± √3, 0); y-int: (0, 12) 27. 34. w = 120ft, l = 120ft, max area: 14,400 ft2 40. −3𝑥 2 + 1 2 23. 𝑓(𝑥) = ( 𝑥) + 3 51. 𝑓 −1 (𝑥) = 54. & 55. See online key. 37. −𝑥 2 + 3𝑥 − 3 42. 9x – 20 1 𝑥 3 52. 𝑓 −1 (𝑥) = √𝑥 − 1 21 ); 4 −3±√21 , 0); 2 x-int: ( 32. 𝐴(𝑤) = 𝑤 2 + 8𝑤 35a) 𝑃(𝑤) = 46. f(x) = ; g(x) = x – 7 3𝑥−1 2 21. 𝑓(𝑥) = (𝑥 + 4)3 − 1 200 + 𝑤 2𝑤 38. 𝑥 2 + 3𝑥 − 7 43. -47 y-int: (0, 3) 33. 𝑃(𝑥) = −𝑥 2 + 50𝑥 35b) w = 10m, l = 10m 39. −3𝑥 3 + 5𝑥 2 + 6𝑥 − 10 44. -2 47. f(x) = √𝑥; g(x) = 1 – x 53. 𝑓 −1 (𝑥) = 𝑥 2 + 2 48. yes 49. no