Chapter 2 Test Review

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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
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