Chapter 10

ⓐ $f\left(4\right)=5$; ⓑ $g\left(f\left(4\right)\right)=32$

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

$x$

$x$

ⓐ $1$; ⓑ $1$

ⓐ $\frac{1}{2}$; ⓑ $3$

$x=9,x=-9$

$\frac{1}{9}$

$x=7$

ⓐ $1$; ⓑ $a$

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

$2.565$

$x=4,x=-4$

$x=2,x=3$

$x=-5,x=1$

ⓐ $15x+1$ ⓑ $15x-9$
ⓒ $15x^2-7x-2$

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

ⓐ –8 ⓑ 5 ⓒ 40

ⓐ 65 ⓑ 10 ⓒ 5

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

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

ⓐ Not a function ⓑ One-to-one function

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

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

Inverse function: $\left\{\left(4,\mathrm{-1}\right),\left(1,\mathrm{-2}\right),\left(0,\mathrm{-3}\right),\left(2,\mathrm{-4}\right)\right\}.$ Domain: $\left\{0,1,2,4\right\}.$ Range: $\left\{\mathrm{-4},\mathrm{-3},\mathrm{-2},\mathrm{-1}\right\}.$

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

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

$f^{\mathrm{-1}}(x)=\frac{x+3}{5}$

$f^{\mathrm{-1}}(x)=\frac{x-5}{8}$

$f^{\mathrm{-1}}(x)=\frac{x^5+2}{3}$

$f^{\mathrm{-1}}(x)=\frac{x^4+7}{6}$

$x=2$

$x=4$

$x=\mathrm{-1},x=2$

$x=\mathrm{-2},x=3$

ⓐ $\text{\$}\mathrm{22,332.96}$
ⓑ $\text{\$}\mathrm{22,362.49}$ ⓒ $\text{\$}\mathrm{22,377.37}$

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

She will find 166 bacteria.

They will find 1,102 viruses.

ⓐ ${\text{log}}_{3}9=2$
ⓑ ${\text{log}}_7\sqrt{7}=\frac{1}{2}$ ⓒ ${\text{log}}_{\frac{1}{3}}\frac{1}{27}=x$

ⓐ ${\text{log}}_{4}64=3$
ⓑ ${\text{log}}_4\sqrt[3]{4}=\frac{1}{3}$ ⓒ ${\text{log}}_{\frac{1}{2}}\frac{1}{32}=x$

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

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

ⓐ $x=8$ ⓑ $x=125$ ⓒ $x=2$

ⓐ
$x=9$ ⓑ $x=243$ ⓒ $x=3$

ⓐ
2 ⓑ $\frac{1}{2}$ ⓒ $\mathrm{-5}$

ⓐ 2 ⓑ $\frac{1}{3}$ ⓒ $\mathrm{-2}$

ⓐ
$a=11$
ⓑ $x=e^7$

ⓐ
$a=4$
ⓑ $x=e^9$

ⓐ
$x=13$
ⓑ $x=2$

ⓐ
$x=6$
ⓑ $x=1$

The quiet dishwashers have a decibel level of 50 dB.

The decibel level of heavy traffic is 90 dB.

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

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

ⓐ 0 ⓑ 1

ⓐ 0 ⓑ 1

ⓐ 15 ⓑ 4

ⓐ 8 ⓑ 15

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

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

ⓐ ${\text{log}}_{4}3-1$ ⓑ $\text{log}x-3$

ⓐ ${\text{log}}_{2}5-2$ ⓑ $1-\text{log}y$

ⓐ $4{\text{log}}_{7}5$ ⓑ $100·\text{log}x$

ⓐ $7{\text{log}}_{2}3$ ⓑ $20·\text{log}x$

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

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

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

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

${\text{log}}_2\frac{5x}{y}$

${\text{log}}_3\frac{6}{xy}$

${\text{log}}_2{x}^3{(x-1)}^2$

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

$3.402$

$2.379$

$x=6$

$x=4$

$x=4$

$x=8$

$x=3$

$x=8$

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

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

$x=\text{ln}9+2\approx 4.197$

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

$r\approx 9.3\text{\%}$

$r\approx 11.9\text{\%}$

There will be 62,500 bacteria.

There will be 47,700 bacteria.

There will be 6.44 mg left.

There will be 31.5 mg left.

ⓐ $8x+23$ ⓑ $8x+11$ ⓒ
$8x^2+26x+15$

ⓐ $24x+1$ ⓑ $24x-19$
ⓒ $24x^2-14x-5$

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

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

ⓐ 245 ⓑ 104 ⓒ 53

ⓐ 250 ⓑ 14 ⓒ 77

Function; not one-to-one

One-to-one function

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

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

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

Inverse function: $\left\{\left(\mathrm{-2},0\right),\left(3,1\right),\left(7,2\right),\left(12,3\right)\right\}.$ Domain: $\left\{\mathrm{-2},3,7,12\right\}.$ Range: $\left\{0,1,2,3\right\}.$

Inverse function: $\left\{\left(\mathrm{-3},\text{−}2\right),\left(\mathrm{-1},\mathrm{-1}\right),\left(1,0\right),\left(3,1\right)\right\}.$ Domain: $\left\{\mathrm{-3},\text{−}1,1,3\right\}.$ Range: $\left\{\mathrm{-2},\mathrm{-1},0,1\right\}.$

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

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

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

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

$f^{\mathrm{-1}}(x)=x+12$

$f^{\mathrm{-1}}(x)=\frac{x}{9}$

$f^{\mathrm{-1}}(x)=6x$

$f^{\mathrm{-1}}(x)=\frac{x+7}{6}$

$f^{\mathrm{-1}}(x)=\frac{x-5}{\mathrm{-2}}$

$f^{\mathrm{-1}}(x)=\sqrt{x-6}$

$f^{\mathrm{-1}}(x)=\sqrt[3]{x+4}$

$f^{\mathrm{-1}}(x)=\frac{1}{x}-2$

$f^{\mathrm{-1}}(x)=x^2+2$, $x\ge 0$

$f^{\mathrm{-1}}(x)=x^3+3$

$f^{\mathrm{-1}}(x)=\frac{x^4+5}{9}$, $x\ge 0$

$f^{\mathrm{-1}}(x)=\frac{x^5-5}{\mathrm{-3}}$

Answers will vary.

$x=4$

$x=\mathrm{-1}$

$x=\mathrm{-1},x=1$

$x=1$

$x=\mathrm{-1}$

$x=2$

$x=\mathrm{-1},x=2$

ⓕ

ⓐ

ⓔ

ⓐ $\text{\$}\mathrm{7,387.28}$ ⓑ $\text{\$}\mathrm{7,434.57}$ ⓒ $\text{\$}\mathrm{7,459.12}$

$\text{\$}\mathrm{36,945.28}$

162 bacteria

288,929,825

Answers will vary.

Answers will vary.

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

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

$\text{log}\frac{1}{100}=\mathrm{-2}$

${\text{log}}_x\sqrt[3]{6}=\frac{1}{3}$

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

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

${\text{log}}_4\frac{1}{64}=\mathrm{-3}$

$\text{ln}x=3$

$64=2^6$

$32=x^5$

$1=7^0$

$9=9^1$

$\mathrm{1,000}={10}^3$

$43=e^x$

$x=11$

$x=4$

$x=125$

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

$x=2$

$x=\mathrm{-2}$

2

0

$\frac{1}{3}$

$\mathrm{-2}$

$\mathrm{-3}$

$\mathrm{-2}$

$a=9$

$a=3$

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

$x=e^4$

$x=5$

$x=17$

$x=6$

$x=3$

$x=\mathrm{-5}\sqrt{5},x=5\sqrt{5}$

$x=\mathrm{-5},x=5$

A whisper has a decibel level of 20 dB.

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

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

Answers will vary.

Answers will vary.

ⓐ 0 ⓑ 1

ⓐ 10 ⓑ 10

ⓐ 15 ⓑ $\mathrm{-4}$

ⓐ $\sqrt{3}$ ⓑ $\mathrm{-1}$

ⓐ 3 ⓑ 7

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

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

$3+\text{log}y$

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

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

$4-\text{log}y$

$4-\text{ln}16$

$5{\text{log}}_{2}x$

$\mathrm{-3}\text{log}x$

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

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

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

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

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

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

$\frac{1}{2}{\text{log}}_3\left(3x+2y^2\right)-{\text{log}}_{3}5-2{\text{log}}_{3}z$

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

2

2

${\text{log}}_2\frac{5}{x-1}$

${\text{log}}_5\frac{2}{xy}$

${\text{log}}_3{x}^6{y}^9$

0

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

$\text{log}{(2x+3)}^2·\sqrt{x+1}$

$2.379$

$1.674$

$5.542$

Answers will vary.

Answers will vary.

$x=7$

$x=4$

$x=1,$ $x=3$

$x=8$

$x=3$

$x=20$

$x=3$

$x=6$

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

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

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

$x=\text{ln}8\approx 2.079$

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

$x=\text{ln}3-2\approx \text{−}0.901$

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

$x=\text{ln}6\approx 1.792$

$x=\text{ln}8+1\approx 3.079$

$x=5$

$x=\mathrm{-4},x=5$

$a=3$

$x=e^9$

$x=7$

$x=3$

$x=2$

$x=6$

$x=5$

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

$x=\text{ln}7-5\approx \text{−}3.054$

$6.9\text{\%}$

13.9 years

122,070 bacteria

8 times as large as the original population

0.03 ml

Answers will vary.

ⓐ $4x^2+12x$ ⓑ $16x^2+12x$ ⓒ $4x^3+12x^2$

ⓐ $\mathrm{-123}$ ⓑ 356 ⓒ 41

Function; not one-to-one

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

Inverse function: $\left\{\left(10,\mathrm{-3}\right),\left(5,\mathrm{-2}\right),\left(2,\mathrm{-1}\right),\left(1,0\right)\right\}.$ Domain: $\left\{1,2,5,10\right\}.$ Range: $\left\{\mathrm{-3},\mathrm{-2},\mathrm{-1},0\right\}.$

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

$f^{\mathrm{-1}}\left(x\right)=\frac{x+11}{6}$

$f^{\mathrm{-1}}\left(x\right)=\frac{1}{x}-5$

$x=\mathrm{-2},x=2$

$x=\mathrm{-1}$

$x=\mathrm{-3},x=5$