13.1.10: Chapter 10

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Be Prepared

10.1

ⓐ $f (4) = 5$ ; ⓑ $g (f (4)) = 32$

10.2

$x = - \frac{2}{3} y + 4$

10.3

$x$

10.4

$x$

10.5

ⓐ $1$ ; ⓑ $1$

10.6

ⓐ $\frac{1}{2}$ ; ⓑ $3$

10.7

$x = 9, x = - 9$

10.8

$\frac{1}{9}$

10.9

$x = 7$

10.10

ⓐ $1$ ; ⓑ $a$

10.11

${(x^{2} y)}^{\frac{1}{3}}$

10.12

$2.565$

10.13

$x = 4, x = - 4$

10.14

$x = 2, x = 3$

10.15

$x = - 5, x = 1$

Try It

10.1

ⓐ $15 x + 1$ ⓑ $15 x - 9$
ⓒ $15 x^{2} - 7 x - 2$

10.2

ⓐ $24 x - 23$ ⓑ $24 x - 23$
ⓒ $24 x^{2} - 38 x + 15$

10.3

ⓐ –8 ⓑ 5 ⓒ 40

10.4

ⓐ 65 ⓑ 10 ⓒ 5

10.5

ⓐ One-to-one function
ⓑ Function; not one-to-one

10.6

ⓐ Not a function
ⓑ Function; not one-to-one

10.7

ⓐ Not a function ⓑ One-to-one function

10.8

ⓐ Function; not one-to-one ⓑ One-to-one function

10.9

Inverse function: ${(4, 0), (7, 1), (10, 2), (13, 3)} .$ Domain: ${4, 7, 10, 13} .$ Range: ${0, 1, 2, 3} .$

10.10

Inverse function: ${(4, −1), (1, −2), (0, −3), (2, −4)} .$ Domain: ${0, 1, 2, 4} .$ Range: ${−4, −3, −2, −1} .$

10.11

$This figure shows a line from (negative 4, negative 3) to (negative 2, negative 2) then to (negative 1, 0) then to (2, 1) and then to (3, 4).$

10.12

$Graph extends from negative 4 to 4 on both axes. Points plotted are (negative 3, 4), (0, 3), (1, 2), and (4, 1). Line segments connect points.$

10.13

$g (f (x)) = x,$ and $f (g (x)) = x,$ so they are inverses.

10.14

$g (f (x)) = x,$ and $f (g (x)) = x,$ so they are inverses.

10.15

$f^{−1} (x) = \frac{x + 3}{5}$

10.16

$f^{−1} (x) = \frac{x - 5}{8}$

10.17

$f^{−1} (x) = \frac{x^{5} + 2}{3}$

10.18

$f^{−1} (x) = \frac{x^{4} + 7}{6}$

10.19 $This figure shows a curve that slopes swiftly upward from just above (negative 3, 0) through (0, 1) up to (1, 4).$

10.20 $This figure shows a curve that slopes swiftly upward from just above (negative 3, 0) through (0, 1) up to (1, 5).$

10.21 $This figure shows a curve that passes through (negative 1, 4), (0, 1) to a point just above (3, 0).$

10.22 $This figure shows a curve that passes through (negative 1, 5), (0, 1) to a point just above (3, 0).$

10.23 $This figure shows the graphs of two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second function g of x equals 2 to the x minus 1 power is marked in red and passes through the points (0, 1 over 2), (1, 1), and (2, 2).$

10.24 $This figure shows the graphs of two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1) and (1, 3). The second function g of x equals 3 to the x plus 1 power is marked in red and passes through the points (negative 2, 1 over 3), (negative 1, 1), and (0, 3).$

10.25 $This figure shows the graphs of two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1) and (1, 3). The second function g of x equals 3 to the x power plus 2 is marked in red and passes through the points (negative 1, 7 over 3), (0, 3) and (1, 5).$

10.26 $This figure shows the graphs of two functions. The first function f of x equals 4 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 4), (0, 1) and (1, 4). The second function g of x equals 4 to the x power minus 2 is marked in red and passes through the points (negative 1, negative 7 over 4), (0, negative 1), and (1, 2).$

10.27

$x = 2$

10.28

$x = 4$

10.29

$x = −1, x = 2$

10.30

$x = −2, x = 3$

10.31

ⓐ $$ 22,332.96$
ⓑ $$ 22,362.49$ ⓒ $$ 22,377.37$

10.32

ⓐ $21,071.81 ⓑ $21,137.04
ⓒ $21,170.00

10.33

She will find 166 bacteria.

10.34

They will find 1,102 viruses.

10.35

ⓐ ${log}_{3} 9 = 2$
ⓑ ${log}_{7} \sqrt{7} = \frac{1}{2}$ ⓒ ${log}_{\frac{1}{3}} \frac{1}{27} = x$

10.36

ⓐ ${log}_{4} 64 = 3$
ⓑ ${log}_{4} \sqrt[3]{4} = \frac{1}{3}$ ⓒ ${log}_{\frac{1}{2}} \frac{1}{32} = x$

10.37

ⓐ $64 = 4^{3}$
ⓑ $1 = x^{0}$ ⓒ $\frac{1}{100} = 10^{−2}$

10.38

ⓐ $27 = 3^{3}$ ⓑ $1 = 3^{0}$
ⓒ $\frac{1}{10} = 10^{−1}$

10.39

ⓐ $x = 8$ ⓑ $x = 125$ ⓒ $x = 2$

10.40

ⓐ
$x = 9$ ⓑ $x = 243$ ⓒ $x = 3$

10.41

ⓐ
2 ⓑ $\frac{1}{2}$ ⓒ $−5$

10.42

ⓐ 2 ⓑ $\frac{1}{3}$ ⓒ $−2$

10.43 $This figure shows the logarithmic curve going through the points (1 over 3, negative 1), (1, 0), and (3, 1).$

10.44 $This figure shows the logarithmic curve going through the points (1 over 5, negative 1), (1, 0), and (5, 1).$

10.45 $This figure shows the logarithmic curve going through the points (1 over 2, 1), (1, 0), and (2, negative 1).$

10.46 $This figure shows the logarithmic curve going through the points (1 over 4, 1), (1, 0), and (4, negative 1).$

10.47

ⓐ
$a = 11$
ⓑ $x = e^{7}$

10.48

ⓐ
$a = 4$
ⓑ $x = e^{9}$

10.49

ⓐ
$x = 13$
ⓑ $x = 2$

10.50

ⓐ
$x = 6$
ⓑ $x = 1$

10.51

The quiet dishwashers have a decibel level of 50 dB.

10.52

The decibel level of heavy traffic is 90 dB.

10.53

The intensity of the 1906 earthquake was about 8 times the intensity of the 1989 earthquake.

10.54

The intensity of the earthquake in Chile was about 1,259 times the intensity of the earthquake in Los Angeles.

10.55

ⓐ 0 ⓑ 1

10.56

ⓐ 0 ⓑ 1

10.57

ⓐ 15 ⓑ 4

10.58

ⓐ 8 ⓑ 15

10.59

ⓐ $1 + {log}_{3} x$
ⓑ $3 + {log}_{2} x + {log}_{2} y$

10.60

ⓐ $1 + {log}_{9} x$
ⓑ $3 + {log}_{3} x + {log}_{3} y$

10.61

ⓐ ${log}_{4} 3 - 1$ ⓑ $log x - 3$

10.62

ⓐ ${log}_{2} 5 - 2$ ⓑ $1 - log y$

10.63

ⓐ $4 {log}_{7} 5$ ⓑ $100 \cdot log x$

10.64

ⓐ $7 {log}_{2} 3$ ⓑ $20 \cdot log x$

10.65

${log}_{2} 5 + 4 {log}_{2} x + 2 {log}_{2} y$

10.66

${log}_{3} 7 + 5 {log}_{3} x + 3 {log}_{3} y$

10.67

$\frac{1}{5} (4 {log}_{4} x - \frac{1}{2} - 3 {log}_{4} y - 2 {log}_{4} z)$

10.68

$\frac{1}{3} (2 {log}_{3} x - {log}_{3} 5 - {log}_{3} y - {log}_{3} z)$

10.69

${log}_{2} \frac{5 x}{y}$

10.70

${log}_{3} \frac{6}{x y}$

10.71

${log}_{2} x^{3} {(x - 1)}^{2}$

10.72

$log x^{2} {(x + 1)}^{2}$

10.73

$3.402$

10.74

$2.379$

10.75

$x = 6$

10.76

$x = 4$

10.77

$x = 4$

10.78

$x = 8$

10.79

$x = 3$

10.80

$x = 8$

10.81

$x = \frac{log 43}{log 7} \approx 1.933$

10.82

$x = \frac{log 98}{log 8} \approx 2.205$

10.83

$x = ln 9 + 2 \approx 4.197$

10.84

$x = \frac{ln 5}{2} \approx 0.805$

10.85

$r \approx 9.3 %$

10.86

$r \approx 11.9 %$

10.87

There will be 62,500 bacteria.

10.88

There will be 47,700 bacteria.

10.89

There will be 6.44 mg left.

10.90

There will be 31.5 mg left.

Section 10.1 Exercises

ⓐ $8 x + 23$ ⓑ $8 x + 11$ ⓒ
$8 x^{2} + 26 x + 15$

ⓐ $24 x + 1$ ⓑ $24 x - 19$
ⓒ $24 x^{2} - 14 x - 5$

ⓐ $6 x^{2} - 9 x$ ⓑ $18 x^{2} - 9 x$
ⓒ $6 x^{3} - 9 x^{2}$

ⓐ $2 x^{2} + 3$ ⓑ $4 x^{2} - 4 x + 3$
ⓒ $2 x^{3} - x^{2} + 4 x - 2$

ⓐ 245 ⓑ 104 ⓒ 53

11.

ⓐ 250 ⓑ 14 ⓒ 77

13.

Function; not one-to-one

15.

One-to-one function

17.

ⓐ Not a function ⓑ Function; not one-to-one

19.

ⓐ One-to-one function
ⓑ Function; not one-to-one

21.

Inverse function: ${(1, 2), (2, 4), (3, 6), (4, 8)} .$ Domain: ${1, 2, 3, 4} .$ Range: ${2, 4, 6, 8} .$

23.

Inverse function: ${(−2, 0), (3, 1), (7, 2), (12, 3)} .$ Domain: ${−2, 3, 7, 12} .$ Range: ${0, 1, 2, 3} .$

25.

Inverse function: ${(−3, − 2), (−1, −1), (1, 0), (3, 1)} .$ Domain: ${−3, − 1, 1, 3} .$ Range: ${−2, −1, 0, 1} .$

27.

$This figure shows a series of line segments from (negative 3, negative 4) to (0, negative 3) then to (2, negative 1), and then to (4, 3).$

29.

$This figure shows a series of line segments from (negative 1, 4) to (2, 3) then to (3, 0), and then to (4, negative 4).$

31.

$g (f (x)) = x,$ and $f (g (x)) = x,$ so they are inverses.

33.

$g (f (x)) = x,$ and $f (g (x)) = x,$ so they are inverses.

35.

$g (f (x)) = x,$ and $f (g (x)) = x,$ so they are inverses.

37.

$g (f (x)) = x,$ and $f (g (x)) = x,$ so they are inverses (for nonnegative $x) .$

39.

$f^{−1} (x) = x + 12$

41.

$f^{−1} (x) = \frac{x}{9}$

43.

$f^{−1} (x) = 6 x$

45.

$f^{−1} (x) = \frac{x + 7}{6}$

47.

$f^{−1} (x) = \frac{x - 5}{−2}$

49.

$f^{−1} (x) = \sqrt{x - 6}$

51.

$f^{−1} (x) = \sqrt[3]{x + 4}$

53.

$f^{−1} (x) = \frac{1}{x} - 2$

55.

$f^{−1} (x) = x^{2} + 2$ , $x \geq 0$

57.

$f^{−1} (x) = x^{3} + 3$

59.

$f^{−1} (x) = \frac{x^{4} + 5}{9}$ , $x \geq 0$

61.

$f^{−1} (x) = \frac{x^{5} - 5}{−3}$

63.

Answers will vary.

Section 10.2 Exercises

65.

$This figure shows a curve that passes through (negative 1, 1 over 2) through (0, 1) to (1, 2).$

67.

$This figure shows a curve that passes through (negative 1, 1 over 6) through (0, 1) to (1, 6).$

69.

$This figure shows a curve that passes through (negative 1, 2 over 3) through (0, 1) to (1, 3 over 2).$

71.

$This figure shows a curve that passes through (negative 1, 2) through (0, 1) to (1, 1 over 2).$

73.

$This figure shows a curve that passes through (negative1, 6) through (0, 1) to (1, 1 over 6).$

75.

$This figure shows a curve that passes through (negative 1, 5 over 2) through (0, 1) to (1, 2 over 5).$

77.

$This figure shows two functions. The first function f of x equals 4 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 4), (0, 1) and (1, 4). The second function g of x equals 4 to the x minus 1 power is marked in red and passes through the points (0, 1 over 4), (1, 1) and (2, 4).$

79.

$This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second function g of x equals 2 to the x minus 2 power is marked in red and passes through the points (0, 1 over 4), (1, 1 over 2), and (2, 1).$

81.

$This figure shows two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1), and (1, 3). The second function g of x equals 3 to the x power plus 2 is marked in red and passes through the points (negative 2, 1), (negative 1, 3), and (0, 5).$

83.

$This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1), and (1, 2). The second function g of x equals 2 to the x power plus 1 is marked in red and passes through the points (negative 1, 1), (0, 2), and (1, 4).$

85.

$This figure shows an exponential curve that passes through (negative 3, 1 over 3), (negative 2, 1), and (0, 9).$

87.

$This figure shows an exponential that passes through (negative 1, 7 over 2), (0, 4), and (1, 5).$

89.

$This figure shows an exponential that passes through (2, 4), (3, 2), and (4, 1).$

91.

$This figure shows an exponential that passes through (1, 1 plus 1 over e), (0, 2), and (1, e).$

93.

$This figure shows an exponential that passes through (negative 1, negative 1 over 2), (0, negative 1), and (1, 2).$

95.

$x = 4$

97.

$x = −1$

99.

$x = −1, x = 1$

101.

$x = 1$

103.

$x = −1$

105.

$x = 2$

107.

$x = −1, x = 2$

109.

ⓕ

111.

ⓐ

113.

ⓔ

115.

ⓐ $$ 7,387.28$ ⓑ $$ 7,434.57$ ⓒ $$ 7,459.12$

117.

$$ 36,945.28$

119.

162 bacteria

121.

288,929,825

123.

Answers will vary.

125.

Answers will vary.

Section 10.3 Exercises

127.

${log}_{2} 32 = 5$

129.

${log}_{5} 125 = 3$

131.

$log \frac{1}{100} = −2$

133.

${log}_{x} \sqrt[3]{6} = \frac{1}{3}$

135.

${log}_{17} \sqrt[5]{17} = x$

137.

${log}_{\frac{1}{3}} \frac{1}{81} = 4$

139.

${log}_{4} \frac{1}{64} = −3$

141.

$ln x = 3$

143.

$64 = 2^{6}$

145.

$32 = x^{5}$

147.

$1 = 7^{0}$

149.

$9 = 9^{1}$

151.

$1,000 = 10^{3}$

153.

$43 = e^{x}$

155.

$x = 11$

157.

$x = 4$

159.

$x = 125$

161.

$x = \frac{1}{243}$

163.

$x = 2$

165.

$x = −2$

167.

169.

171.

$\frac{1}{3}$

173.

$−2$

175.

$−3$

177.

$−2$

179.

$This figure shows the logarithmic curve going through the points (1 over 4, negative 1), (1, 0), and (4, 1).$

181.

$This figure shows that the logarithmic curve going through the points (1 over 7, negative 1), (1, 0), and (7, 1).$

183.

$This figure shows the logarithmic curve going through the points (2 over 5, negative 1), (1, 0), and (2.5, 1).$

185.

$This figure shows the logarithmic curve going through the points (1 over 5, 1), (1, 0), and (5, negative 1).$

187.

$This figure shows the logarithmic curve going through the points (3 over 5, 1), (1, 0), and (5 over 3, negative 1).$

189.

$a = 9$

191.

$a = 3$

193.

$a = 2 \sqrt[3]{3}$

195.

$x = e^{4}$

197.

$x = 5$

199.

$x = 17$

201.

$x = 6$

203.

$x = 3$

205.

$x = −5 \sqrt{5}, x = 5 \sqrt{5}$

207.

$x = −5, x = 5$

209.

A whisper has a decibel level of 20 dB.

211.

The sound of a garbage disposal has a decibel level of 100 dB.

213.

The intensity of the 1994 Northridge earthquake in the Los Angeles area was about 40 times the intensity of the 2014 earthquake.

215.

Answers will vary.

217.

Answers will vary.

Section 10.4 Exercises

219.

ⓐ 0 ⓑ 1

221.

ⓐ 10 ⓑ 10

223.

ⓐ 15 ⓑ $−4$

225.

ⓐ $\sqrt{3}$ ⓑ $−1$

227.

ⓐ 3 ⓑ 7

229.

${log}_{5} 8 + {log}_{5} y$

231.

$4 + {log}_{3} x + {log}_{3} y$

233.

$3 + log y$

235.

${log}_{6} 5 - 1$

237.

$3 - {log}_{5} x$

239.

$4 - log y$

241.

$4 - ln 16$

243.

$5 {log}_{2} x$

245.

$−3 log x$

247.

$\frac{1}{3} {log}_{5} x$

249.

$\sqrt[3]{4} ln x$

251.

${log}_{2} 3 + 5 {log}_{2} x + 3 {log}_{2} y$

253.

$\frac{1}{4} {log}_{5} 21 + 3 {log}_{5} y$

255.

${log}_{5} 4 + {log}_{5} a + 3 {log}_{5} b$
$+ 4 {log}_{5} c - 2 {log}_{5} d$

257.

$\frac{2}{3} {log}_{3} x - 3 - 4 {log}_{3} y$

259.

$\frac{1}{2} {log}_{3} (3 x + 2 y^{2}) - {log}_{3} 5 - 2 {log}_{3} z$

261.

$\frac{1}{3} ({log}_{5} 3 + 2 {log}_{5} x - {log}_{5} 4$
$- 3 {log}_{5} y - {log}_{5} z)$

263.

265.

267.

${log}_{2} \frac{5}{x - 1}$

269.

${log}_{5} \frac{2}{x y}$

271.

${log}_{3} x^{6} y^{9}$

273.

275.

$ln \frac{x^{3} y^{4}}{z^{2}}$

277.

$log {(2 x + 3)}^{2} \cdot \sqrt{x + 1}$

279.

$2.379$

281.

$1.674$

283.

$5.542$

285.

Answers will vary.

287.

Answers will vary.

Section 10.5 Exercises

289.

$x = 7$

291.

$x = 4$

293.

$x = 1,$ $x = 3$

295.

$x = 8$

297.

$x = 3$

299.

$x = 20$

301.

$x = 3$

303.

$x = 6$

305.

$x = \frac{5}{3}$

307.

$x = \frac{log 74}{log 2} \approx 6.209$

309.

$x = \frac{log 112}{log 4} \approx 3.404$

311.

$x = ln 8 \approx 2.079$

313.

$x = \frac{log 8}{log \frac{1}{3}} \approx − 1.893$

315.

$x = ln 3 - 2 \approx − 0.901$

317.

$x = \frac{ln 16}{3} \approx 0.924$

319.

$x = ln 6 \approx 1.792$

321.

$x = ln 8 + 1 \approx 3.079$

323.

$x = 5$

325.

$x = −4, x = 5$

327.

$a = 3$

329.

$x = e^{9}$

331.

$x = 7$

333.

$x = 3$

335.

$x = 2$

337.

$x = 6$

339.

$x = 5$

341.

$x = \frac{log 10}{log \frac{1}{2}} \approx − 3.322$

343.

$x = ln 7 - 5 \approx − 3.054$

345.

$6.9 %$

347.

13.9 years

349.

122,070 bacteria

351.

8 times as large as the original population

353.

0.03 ml

355.

Answers will vary.

Review Exercises

357.

359.

361.

Function; not one-to-one

363.

ⓐ Function; not one-to-one ⓑ Not a function

365.

Inverse function: ${(10, −3), (5, −2), (2, −1), (1, 0)} .$ Domain: ${1, 2, 5, 10} .$ Range: ${−3, −2, −1, 0} .$

367.

$g (f (x)) = x,$ and $f (g (x)) = x,$ so they are inverses.

369.

$f^{−1} (x) = \frac{x + 11}{6}$

371.

$f^{−1} (x) = \frac{1}{x} - 5$

373.

$This figure shows an exponential line passing through the points (negative 1, 1 over 4), (0, 1), and (1, 4).$

375.

$This figure shows an exponential line passing through the points (negative 1, 4 over 3), (0, 1), and (1, 3 over 4).$

377.

$This figure shows an exponential line passing through the points (negative 1, negative 59 over 23), (0, negative 2), and (1, negative7 over 10).$

379.

$This figure shows an exponential line passing through the points (negative 1, negative 1 over e), (0, negative 1), and (1, negative e).$

381.

$x = −2, x = 2$

383.

$x = −1$

385.

$x = −3, x = 5$

387.

$$ 163,323.40$

389.

330,259,000

391.

$log \frac{1}{1,000} = −3$

393.

$ln 16 = y$

395.

$100000 = 10^{5}$

397.

$x = 5$

399.

$x = 4$

401.

403.

$This figure shows a logarithmic line passing through the points (1 over 5, negative 1), (1, 0), and (5, 1).$

405.

$This figure shows a logarithmic line passing through the points (4 over 5, 1), (1, 0), and (5 over 4, negative 1).$

407.

$x = e^{−3}$

409.

$x = 8$

411.

90 dB

413.

ⓐ 13 ⓑ $−9$

415.

ⓐ 8 ⓑ 5

417.

$4 + log m$

419.

$5 - ln 2$

421.

$\frac{1}{7} {log}_{4} z$

423.

${log}_{5} 8 + 2 {log}_{5} a + 6 {log}_{5} b$
$+ {log}_{5} c - 3 {log}_{5} d$

425.

$\frac{1}{3} ({log}_{6} 7 + 2 {log}_{6} x - 1 - 3 {log}_{6} y$
$- 5 {log}_{6} z)$

427.

${log}_{3} x^{3} y^{7}$

429.

$log \frac{\sqrt[4]{y}}{{(y - 3)}^{2}}$

431.

5.047

433.

$x = 4$

435.

$x = 3$

437.

$x = \frac{log 101}{log 2} \approx 6.658$

439.

$x = \frac{log 7}{log \frac{1}{3}} \approx − 1.771$

441.

$x = ln 15 + 4 \approx 6.708$

443.

11.6 years

445.

12.7 months

Practice Test

447.

449.

ⓐ Not a function ⓑ One-to-one function

451.

$f^{−1} (x) = \sqrt[5]{x + 9}$

453.

$x = 5$

455.

457.

$343 = 7^{3}$

459.

461. $This figure shows a logarithmic line passing through (1 over 3, 1), (1, 0), and (3, 1).$

463.

40 dB

465.

$2 + {log}_{5} a + {log}_{5} b$

467.

$\frac{1}{4} ({log}_{2} 5 + 3 {log}_{2} x - 4 - 2 {log}_{2} y$
$- 7 {log}_{2} z)$

469.

$log \frac{\sqrt[6]{x}}{{(x + 5)}^{3}}$

471.

$x = 6$

473.

$x = ln 8 + 4 \approx 6.079$

475.

1,921 bacteria

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Section 10.1 Exercises

Section 10.2 Exercises

Section 10.3 Exercises

Section 10.4 Exercises

Section 10.5 Exercises

Review Exercises

Practice Test