Algebra 2 Final Exam Review Answers 1. Evaluate: 1252 / 3 = ( 3 125) 2 2. Evaluate. 165/ 4 = ( 4 16)5
52 25
25 32
5
3. Rewrite y 7 using radical notation. ( 7 y )5 Simplify: 4. x
13
14
x = x
1 1 3 4
x
4 3 12 12
5. (a5b6c10 )4 = a 5 4b 6 4 c10 4 6.
15 x 4 y 7 5 x5 y 8 z 2
15 x 5
4 5
y7
x
7 12
a 20b 24c 40 8 0
z
2
3x 9 y15 z 2
3 y15 z 2 x9
7. Find an equation for the inverse of the relation y 5x 3. x 5y 3
x 3 5y x 3 5
y
8. Let f ( x ) – 4 x . Find f 1 . x 4y x y f 1 4 9. If f(x) = 4x2 + 7 and g(x) = 5x + 2. Find g(f(x)). g (4 x 2 7) 5(4 x 2 7) 2
20 x 2 35 2 20 x 2 37
10. Find f(g(x))? What does your answer mean? 1 1 6x 2 g( x) x , f ( x) 2 3 3 1 1 6 x 2 1 1 2 3 f x 2 3 3 3x 2 2 Since f(g(x)) = x, this means f(x) and g(x) are inverses of each other. 3 3x 3 x
11. If f(x) = x3 + 2x2 – x + 4 find f(2) f(2) = 23+2(2)2 – 2 + 4 =8 + 8 – 2 + 4 = 18 12. Sketch the graph of the function and its inverse on the same coordinate plane. y
x 2 3
Pick out points on the actual graph. Switch the (x, y) points. Graph those. So actual graph has points at (2, 3) and (3, 4). Inverse has points at (3, 2) and (4, 3). Graph these points.
13. Describe how to obtain the graph of y Moved 3 to the left and down 3 Graph #14 and 15: 14. f(x) = x 5
Graph moves up 5
Refer to the function g ( x )
2
x 1.
16. Sketch the graph of g(x).
17. What is the domain of g(x) above? [– 1, ∞)
x 3 3 from the graph of y
15. f(x) =
3
x 3
5
x.
Graph moves right 3 and down 5
18. Refer to the function g ( x ) 1 [1, ∞)
x 3 . What is the range of g(x)?
Solve the equation. Check for extraneous solutions. 19. x + 2 = x x 2 x2
x2
0
x 2
But x = – 1 is extraneous, so the only answer is x = 2
0 ( x 2)( x 1) x 20.
x
2, x
1
56 = x x 56 x 2 0
x2
x 56
But x = – 7 is extraneous, so the only answer is x = 8 [A]
0 ( x 8)( x 7) x 8, x [A] 8 21. Solve
3
x 5 x 5 x
7 [B] no solution
[C] 8, –7
[D] –7
–4 64 59
22. Find the mean of the set of numbers, to the nearest hundredth. (by hand!) 24, 11, 40, 29, 21, 11, 33, 11 Mean = (24+11+40+29+21+11+33+11)/8 = 22.5 23. Find the median of the set of numbers. 13, 22, 26, 4, 33, 3, 21, 18, 34 Median = # listed in middle when in order: 3, 4, 13, 18, 21, 22, 26, 33, 34 [A] 21 [B] 23.8 [C] 31 [D] 19.3 24. Find the mode of the set of data. 10, 18, 19, 13, 18, 19, 10, 13, 19, 12 Mode = most common number [A] 19 [B] 15.5 [C] 17 25. Graph: f(x) = 3x + 2
List the domain, range and any asymptotes. Domain: (– ∞, ∞) Range: (2, ∞) Horizontal Asymptote: y = 2
[D] 15.1
26. Graph f(x) = ex
List domain, range and any asymptotes. Domain: (– ∞, ∞) Range: (0, ∞) Horizontal Asymptote: y = 0
27.
Set up the equation to find the value of $1000 deposited for 10 years in an account paying 6% annual interest compounded monthly. nt r A P 1 n 0.06 1000 1 12 1819.40
12 10
28. How much money must be deposited now in an account paying 8% annual interest, compounded quarterly, to have a balance of $1000 after 10 years? (set up the equation and solve up until the point you need a calculator) nt r A P 1 n 0.08 1000 1 4 1000 1 0.02
4 10
40
1000(1.02) 40
Graph: 29. f ( x )
F 1I 2G J H 4K
x
State the domain and range for this graph
Domain: (– ∞, ∞) Range: (0, ∞) Horizontal Asymptote: y = 0
Graph: 30. f ( x) ln x
31. State the domain and range for the graph above. Domain: (0, ∞) Range: (– ∞, ∞) (e 2 x ) 2 10e x e4 x 5e 2 x 4 x 10e x e4 x x 5e 2 x 4 x 10 5x e 5e 6 x 10 1 2e x 3 in exponential form. log 243 27 5
32. Simplify the expression.
33. Write the equation
5e 2 x e4 x
243
3 5
27
34. Evaluate: log 3 9 2 since 32 9 23x – 1 = 16 23 x 1 2 4 3x 1 4 3x 5 5 x 3 36. Find the inverse of the function. y x 35. Solve for x:
8x
log8 x log 8 y
y
37. Evaluate without using a calculator. log 2 16
4 since 24
16
38. Evaluate the expression. log1 5 125
1 5
?
125 ?
3
39. Graph the function. State the domain and range. y
ln x 4
Domain: (– ∞, ∞) Range: (0, ∞) Horizontal Asymptote: y = 0
Use the change-of-base formula to evaluate the expression (40 and 41) (just show set up) 40. log 4 7 log 7 log 4
41. log 4 24 log 24 log 4
42. Express as a single logarithm: log a 11 log a 35 log a 11 35 log a 385
43. Condense the expression.
log 5 16
1 log516 3log5 x 4 log5 y 2 1
2
log 5 x3 log 5 y 4
log 5 4 log 5 x3 log 5 y 4 log 5
4 y4 x3
44. Solve:
1 = 27 3 x 9
3
2
33
3
2
39 x
5
3x 5
15
2 9 x 15 13 9 x 13 9 13 [A] 9 x
[B]
1 3
[C] 1
45. Sketch the graph of the function. f ( x )
46. Sketch the graph of the function f(x) =
[D]
17 9
x 2 x 2
2 x 4
3 . Be sure to label all asymptotes.
Horizontal asymptote: y = 3 Vertical asymptote: x = – 4
47. Identify all horizontal and vertical asymptotes of the graph of the function. f ( x ) Horizontal: y = 0 Vertical: x = 1, x = – 1 48. Simplify the rational expression.
n 2 8n 15 n 2 25 ( n 3)( n 5) ( n 5)( n 5) n 3 n 5
5x x
2
1
8 12 2x 5 x 4 8( x 4) 12(2 x 5)
49. Solve:
8 x 32
24 x 60
92 16 x 23 4
x
50. Solve the equation. x 30
30 x
x 30
1 5x
1 6 LCD = 30, so multiply every term by 30x 1 1 30 x 30 x 5x 6 x2 6 5x
x2 5x 6 0 ( x 6)( x 1) 0 x
51. Solve the equation.
2x x 2
6, x 1
x
2
1 1
4
LCD = x2 – 4, so multiply every term by the x2 – 4
2x 1 x2 4 2 1 x2 4 x 2 x 4 2 2 x( x 2) 1 x 4
x2 4
2x2 x2
4x
x2 3
4x 3 0
( x 3)( x 1) 0 x
3, x
1
2
52. Rewrite 5x 3 using radical notation.
NOT 3 5x2 !
5 3 x2 53. Solve log5 x 5
log 5 x
x
5
2 2
25
54. The mean age of a person starting to work at Yogurtland is 20. The standard deviation is 2. What percent of workers are between the ages of 18 and 24?
18 to 24 = 34%+34%+13.5% = 81.5% 14 16 18 20 22 24 26
55. What has the larger standard deviation: [a] the average age of the students in the sophomore class [b] the average age of the students in the entire school (numbers are more spread out)
