"

Other Functions

Prep for Polynomial Functions and Their Graphs

      1. two
      2. [latex](-2,0)[/latex] and [latex](2,0)[/latex]
      3. cross
      1. three
      2. [latex](-3,0)[/latex], [latex](-1,0)[/latex], and [latex](2,0)[/latex]
      3. cross
      1. one
      2. [latex](-2,0)[/latex]
      3. bounces off
      1. none
      2. none
      3. neither
    2. [latex]x=-\frac12\;\text{and}\;x=3[/latex]
    3. [latex]\begin{array}{lc}x=-2&x=9\\x+2=0&x-9=0\\(x+2)(x-9)=0&{}\end{array}[/latex]
    1. [latex](x+3)(x-8) = 0[/latex]
    2. [latex]x(x-5) = 0[/latex]
    3. [latex](x-4)(10x+7) = 0[/latex]
    4. [latex](x-2)^2 = 0[/latex]
    5. [latex](3x-2)(2x+1) = 0[/latex]
    6. [latex](x+4)(5x-3) = 0[/latex]

Polynomial Functions and Their Graphs

    1. 3 roots
    2. 4th degree
    3. [latex]f(x) = x(x-1)^2(x-2)[/latex]
    1. 4 roots
    2. 4th degree
    3. [latex]f(x) = (x-1)(x-2)(x-3)(x-4)[/latex]
    1. 2 roots
    2. 3th degree
    3. [latex]f(x) = x(x+2)^2[/latex]
    1. 2 roots
    2. 3th degree
    3. [latex]f(x) = (2x+1)^2(x-3)[/latex]

Homework: Polynomial Functions and Their Graphs

    1. i
    2. c
    3. m
    4. b
    5. a
    6. e
    7. l
    8. f
    9. d
    10. n
    11. j
    12. g
    13. h
    14. k

Prep for Rational Functions

    1. undefined
    2. 0
    3. undefined
    4. 0
    1. [latex]x = 7[/latex]
    2. [latex]x = -3[/latex]
    3. [latex]x = \frac{9}{2}[/latex]
    4. [latex]x = -\frac{7}{4}[/latex]
    5. [latex]x = -1[/latex]
    6. [latex]x = 1, -\frac{1}{2}[/latex]
    7. [latex]x = -7, \frac{3}{2}[/latex]
      2. undefined
      3. [latex](2,0)[/latex]
      4. none
      2. [latex]m=0[/latex]
      3. none
      4. [latex](0,-3)[/latex]
    2. (a), (c), (e), (i), (j)

Rational Functions

x f(x) x f(x)
1 1 1 1
0.5 2 2 0.5
0.25 4 4 0.25
0.2 5 5 0.2
0.125 8 8 0.125
0.1111 [latex]\approx9[/latex] 9 [latex]\approx0.1111[/latex]
0.1 10 10 0.1
  1. The value of y becomes larger as x becomes smaller.
  2. The value of y becomes smaller as x becomes larger.
    1. [latex]VA:\;x\;=\;6;\;\text{Domain}:\;x\neq\;6\;\text{or}\;(\infty,\;6)\;\cup\;(6,\;\infty)[/latex]
    2. [latex]VA:\;x\;=\;-2,8;\;\text{Domain}:\;x\neq\;-2,8\;\text{or}\;(-\infty,\;-2)\;\cup\;\left(-2,8\right)\cup(8,\;\infty)[/latex]
    3. [latex]VA:\;x=\pm2;\;\text{Domain}:x\neq\pm2;\;\text{or}\;(-\infty,-2)\;\cup\;\left(-2,2\right)\cup(2,\;\infty)[/latex]
    4. [latex]VA:\;x=0,8;\;\text{Domain}:x\neq0,8\;\text{or}\;(-\infty,0)\;\cup\;\left(0,8\right)\cup(8,\;\infty)[/latex]
    5. Many answers possible.
    6. Answers may vary.
    7. Many answers possible.
      1. 1,800 flowers
      2. 90,000 flowers
      3. 100% removal is impossible because it requires infinitely many flowers.
      1. $2,333
      2. The cost of removing 2 tons of pollution would be infinite.
      1. The more you play, the lower your score.
      2. Any score less than or equal to 18

Homework: Rational Functions

    1. [latex]\begin{array}{l}\text{VA:}x=\frac{10}3\text{; Domain:}x\neq\frac{10}3 \text{ or}\\\left(-\infty,\frac{10}3\right)\cup\left(\frac{10}3,\infty\right)\end{array}[/latex]
    2. [latex]\begin{array}{l}\text{VA:}x=\pm4\text{; Domain:}x\neq\pm4 \text{ or}\\{(-\infty,-4)}\cup{(-4,4)}\cup{(4,\infty)}\end{array}[/latex]
    3. [latex]\begin{array}{l}\text{VA:}x=0\text{; Domain:}x\neq0 \text{ or}\\{(-\infty,0)}\cup{(0,\infty)}\end{array}[/latex]
    4. [latex]\begin{array}{l}\text{VA:}x=-1,5\text{; Domain:}x\neq-1,5 \text{ or}\\{(-\infty,-1)}\cup{(-1,5)}\cup{(5,\infty)}\end{array}[/latex]
    5. Answers may vary.
    6. Many answers are possible.
    7. Explain.
    8. Explain.
      1. [latex]t=\frac dr[/latex]
      2. 3 hours, 15 min
      3. Time approaches infinity
      1. $445,000,000
      2. 1,954 miles because cost would be infinite.
      1. [latex]P=\frac{n\left(8.314\right)T}V[/latex]
      2. When volume increases, pressure decreases.
      3. When volume decreases, pressure increases.
      4. V could not be 0 because pressure would be infinite.
      1. 6 liters
      2. [latex]\;325.16\;K\;\text{or}\;52.16^\circ C[/latex]
      3. 1329.04 in3
      4. [latex]35.57^\circ C[/latex]

Prep for Exponential Functions

    1. 0.65
    2. 0.105
    3. 0.0025
    4. 0.0875
    5. 0.000045
    6. 0.076
    7. -0.00035
    1. [latex]\begin{array}{lc}D&x^3\\F&3x\\B&\frac13x\\C&x-3\\E&x+3\\A&3^x\end{array}[/latex]
    2. They are not the same.
      1. 1.07%, 0.0107
      2. $500
      3. 5 years
    2. compounded weekly
      1. [latex]1\frac14\%\;1.25\%,\;0.0125[/latex]
      2. 12 months, 1 year
      1. 2.5%, 0.025
      2. 15 months, 1.25 years

Exponential Functions

    1. Domain: [latex]\left(-\infty,\infty\right)[/latex]
      Range: [latex]\left(0,\infty\right)[/latex]
      y-intercept: [latex](0,1)[/latex]
    2. Domain: [latex]\left(-\infty,\infty\right)[/latex]
      Range: [latex]\left(0,\infty\right)[/latex]
      y-intercept: [latex](0,1)[/latex]
    3. Domain: [latex]\left(-\infty,\infty\right)[/latex]
      Range: [latex]\left(0,\infty\right)[/latex]
      y-intercept: [latex](0,1)[/latex]
      1. [latex]$130[/latex]
      2. [latex]$134.59[/latex]
      3. [latex]$134.95[/latex]
      4. [latex]$134.98[/latex]
      5. [latex]$134.99[/latex]
      1. [latex]$900.00[/latex]
      2. [latex]$905.13[/latex]
      3. [latex]$906.40[/latex]
      4. [latex]$906.49[/latex]
      5. [latex]$906.52[/latex]
      6. [latex]$906.52[/latex]
      1. 25,000 bacteria
      2. 83,003 bacteria
    2. [latex]$12,754.31[/latex]
    3. [latex]$44,260.17[/latex]
    4. [latex]$227,726.54[/latex]
    5. Bank B
      1. 0.972 kg
      2. 0.752 kg
      1. 10 mg
      2. 8.41 mg at 1 hour; 5.01 mg at 4 hours

Homework: Exponential Functions

    1. Domain: [latex]\left(-\infty,\infty\right)[/latex]
      Range: [latex]\left(0,\infty\right)[/latex]
      y-intercept: [latex](0,1)[/latex]
    2. Domain: [latex]\left(-\infty,\infty\right)[/latex]
      Range: [latex]\left(0,\infty\right)[/latex]
      y-intercept: [latex](0,1)[/latex]
    3. Explain.
    4. Explain.
    1. [latex]$9,449.01[/latex]
    2. [latex]$2,139.05[/latex]
    3. [latex]$2,208.04[/latex]
    4. [latex]$18,505.17[/latex]
    5. [latex]$618,990.79[/latex]
    6. The second loan
      1. [latex]9.42\times10^{-12}[/latex] thousand TBq
      2. 0 TBq
      3. Answer will vary each year
    7. 3.35 mg at 1 hour; 0.45 mg at 6 hours

Prep for Inverse Functions

    1. x y = 2x
      -2 -4
      -1 -2
      0 0
      1 2
      2 4
      3 6
    2. x [latex]y=\frac{x}{2}[/latex]
      -4 -2
      -2 -1
      0 0
      2 1
      4 2
      6 3

    3. The x-and y-coordinates are switched.
    1. x
    2. x
    3. x
    4. x
    5. 4
    6. 6
    7. 2
    8. 8
    1. 0.45
    2. 0.1825
    3. 0.0015
    4. 0.129
    5. 0.00005
    6. -0.022
    7. -0.00269
      1. [latex]$7,000[/latex]
      2. [latex]8.5\%, 0.085[/latex]
      3. [latex]$10,000[/latex]
      4. time
      1. The balance is [latex]$14,000[/latex] instead of [latex]$10,000[/latex]
      2. time
      1. [latex]$1,000[/latex]
      2. [latex]$1,245[/latex]
      3. interest rate

Inverse Functions

    1. x [latex]y = x^2[/latex]
      0 0
      [latex]\frac{1}{3}[/latex] [latex]\frac{1}{9}[/latex]
      [latex]\frac{1}{2}[/latex] [latex]\frac{1}{4}[/latex]
      1 1
      2 4
      3 9
    2. x [latex]y = x^{1/2}[/latex]
      0 0
      [latex]\frac{1}{9}[/latex] [latex]\frac{1}{3}[/latex]
      [latex]\frac{1}{4}[/latex] [latex]\frac{1}{2}[/latex]
      1 1
      4 2
      9 3

    3. The x- and y-coordinates are switched.
    1. [latex]x=9[/latex]
    2. [latex]x=3[/latex]

Inverse Functions — In (x) and [latex]e^x[/latex]

    1. x [latex]y = e^x[/latex]
      -3 0.05
      -2 0.14
      -1 0.37
      0 1
      1 2.72
      2 7.39
    2. x [latex]y = \ln(x)[/latex]
      0.05 -3
      0.14 -2
      0.37 -1
      1 0
      2.72 1
      7.39 2

    3. The x- and y- coordinates are switched.
      [latex]y=e^x[/latex], Domain: [latex]\left(-\infty,\infty\right)[/latex] Range: [latex]\left(0,\infty\right)[/latex]
      y = ln(x), Domain: [latex]\left(0,\infty\right)[/latex] Range: [latex]\left(-\infty,\infty\right)[/latex].
    1. 0
    2. [latex]\approx2.08[/latex]
    3. [latex]\approx6.68[/latex]
    4. [latex]\approx8.99[/latex]
    5. Explain in your own words.
    1. 3
    2. 13
    3. 7
    4. 5
    5. 8
    6. 15
    1. [latex]k = \frac{\ln(8)}{10}[/latex]
    2. [latex]x = \frac{1}{2}[/latex]
    3. [latex]x = e + 2[/latex]
    4. [latex]x = 3[/latex]
    5. [latex]P = \frac{50}{e^{0.6}}[/latex]
    6. [latex]k = \frac{\ln(\frac{1}{2})}{7}[/latex]
    1. [latex]t = 23.1 \text{ years}[/latex]
    2. [latex]10.62\% \text{ per year}[/latex]
    3. [latex]P = \$2,137.07[/latex]

Homework: Inverse Functions — ln(x) and [latex]e^x[/latex]

    1. [latex]x = \frac{\ln(12)}{3}[/latex]
    2. [latex]x = \ln(72)[/latex]
    3. [latex]t = \frac{\ln(5)}{.06}[/latex]
    4. [latex]t=-\frac{\ln\left(\frac12\right)}{.0000124}[/latex]
    5. [latex]r = \frac{\ln(\frac{1}{2})}{15}[/latex]
    6. [latex]x = e^{3.548}[/latex]
    7. [latex]x = \frac{e^5 - 1}{3}[/latex]
    8. [latex]x = \frac{e + 7}{4}[/latex]
    9. [latex]r = \frac{13}{150}[/latex]
    10. [latex]I = 24[/latex]
    11. Answers may vary.
    12. Explain.
    1. 4.2 years (or) 4 years, 2 months, and 12 days
    2. 8.15 years
      1. 11.55 years
      2. 18.31 years
      3. [latex]$725.00[/latex]
      4. 4% per year
      5. 5.15% per year

Inverse Functions — log(x) and [latex]10^x[/latex]

    1. x [latex]y = 10^x[/latex]
      -3 0.001
      -2 0.01
      -1 0.1
      0 1
      1 10
      2 100
    2. x y
      0.001 -3
      0.01 -2
      0.1 -1
      1 0
      10 1
      100 2

    3. The x- and y- coordinates are switched.[latex]y=10^x[/latex], Domain: [latex]\left(-\infty,\infty\right)[/latex] Range: [latex]\left(0,\infty\right)[/latex].[latex]y=\log\left(x\right)[/latex], Domain: [latex]\left(0,\infty\right)[/latex] Range: [latex]\left(-\infty,\infty\right)[/latex].
    1. 0
    2. [latex]\approx0.903[/latex]
    3. [latex]\approx2.903[/latex]
    4. [latex]4[/latex]
    5. Explain in your own words.
    1. 5
    2. 9
    3. 7
    4. 2
    5. 5
    1. [latex]x = \log(400)[/latex]
    2. [latex]x = 3[/latex]
    3. [latex]x = \frac{1}{2}[/latex]
    4. [latex]x = 12[/latex]
    5. [latex]x = \log(12)[/latex]
    6. [latex]x = 10^{-7}$ or $0.0000001[/latex]
    1. Answers will vary.
    2. [latex]k=-1.93%[/latex] per hour
    3. [latex]\approx3.04\;\mathrm{pH}[/latex]
    4. [latex]3.16\times10^{-12}\;\mathrm{mol}/\mathrm L[/latex]
    5. [latex]\approx2.4\;\mathrm{pH}[/latex]
    6. No, both have a [latex]\mathrm{pH}<5[/latex]
    7. [latex]\approx\;7.38[/latex] magnitude
    8. [latex]1.26\times10^6\;\mathrm{mm}[/latex]

Homework: Inverse Functions — [latex]\log\left(x\right)[/latex] and [latex]10^x[/latex]

    1. [latex]x = \log(30)[/latex]
    2. [latex]x = 2[/latex]
    3. [latex]x = \log(3)[/latex]
    4. [latex]x = 102[/latex]
    5. [latex]x = \frac{1}{3}[/latex]
    6. [latex]P \approx \$3,785.74[/latex]
    7. [latex]x = \frac{\log(20)}{0.05}[/latex]
    8. [latex]x=\frac{\log\left(\frac32\right)}3[/latex]
      1. 1.37% per year
      2. 1.75 million people
      3. [latex]\approx29.5[/latex] years after 2010, or sometime in 2039
    2. 6 days
    3. 34.66% per min
    4. -0.04% per year; the sign is negative because the function models decay
    5. 2 hours
    6. [latex]4.0\times10^{-8}\;\mathrm{mol}/\mathrm L[/latex]
    7. [latex]4.7\;\mathrm{pH}[/latex]
    8. [latex]3.16\times10^{-4}\;\mathrm{mol}/\mathrm L[/latex]
    9. [latex]12.4\;\mathrm{pH}[/latex]
    10. 7.09 magnitude
    11. 63.1 mm
    12. 630.96 mm
    13. 100 times stronger
    14. 6.699 magnitude; 10.699 magnitude; every power of 10 increase/decrease in seismographic reading results in a corresponding 1 point increase/decrease in Richter scale reading.

Solving Other Exponential and Logarithmic Equations

    1. [latex]x = 9[/latex]
    2. [latex]x = -2[/latex]
    3. [latex]x = 218[/latex]
    4. [latex]x = \sqrt{7} + 2[/latex]
    5. [latex]x = -\frac{11}{4}[/latex]
    6. [latex]x = 38[/latex]
    1. Take ln of both sides.
    2. Make both sides exponents of base 10.
    3. Make both sides exponents of base 3.
    1. [latex]3 \ln x[/latex]
    2. [latex]5 \log(x-2)[/latex]
    3. [latex]-\frac{1}{2} \log_5 x[/latex]
    4. [latex]\frac{2}{5} \ln x[/latex]
    5. [latex]c \ln b[/latex]
    1. 9.551
    2. 22.925
    3. 69.661
    4. 0.28125
    5. 40
    6. 2.46875
    1. [latex]x = -\frac{11}{2}[/latex]
    2. [latex]x = 3[/latex]
    3. [latex]x = 3[/latex]
    4. [latex]x = 4[/latex]
    5. [latex]x = 2[/latex]
    6. [latex]x = \frac{3}{5}[/latex]
    7. [latex]x = -2[/latex]
    8. [latex]x = -\frac{3}{2}[/latex]

Homework: Solving Other Exponential and Logarithmic Equations

    1. [latex]x=\log(34,926)[/latex]
    2. [latex]x=\ln(64)[/latex]
    3. [latex]x = 2,187[/latex]
    4. [latex]x=\log_4(1024)=5[/latex]
    5. [latex]x=\frac{e^3}5[/latex]
    6. [latex]x=\frac{\ln(11250)+5}4[/latex]
    7. [latex]x=\frac{\ln(7957)-1}{-8}[/latex]
    8. [latex]x=\frac{\ln\left(\frac{10273}4\right)}7[/latex]
    9. [latex]x = \frac{3}{2}[/latex]
    10. [latex]x = 5[/latex]
    11. [latex]x = 4[/latex]
    12. [latex]t=\frac{\ln(5)}{0.06}[/latex]
    13. [latex]k=\frac{\ln\left(\frac12\right)}7[/latex]
    14. [latex]k=\frac{\ln\left(\frac12\right)}{10,000}[/latex]
    15. [latex]t = \frac{\ln(2)}{\ln(1.01)}[/latex]
    16. [latex]x = 25[/latex]
    17. [latex]x = \frac{2}{3}[/latex]
    18. [latex]x = \frac{\log_4(6)}{3}[/latex]
    19. [latex]x = 47.5[/latex]
    20. [latex]x = 200[/latex]
    1. 4% per year
    2. 3.4 years
      1. 3,351 pop
      2. 4,492 pop
      3. 5,679 pop
    3. [latex]P\;=\;240,\;360e^{.012t}[/latex]; 11.22 years from 2000
    4. 158.64% per min
      1. 0.028 per year (decay)
      2. 2,485 years
    5. 4% per year
    6. 3.75% per year
    7. $8,824.97
    8. FYAB is better
      1. 7.5 min
      2. 66.17 bmp

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College Algebra for Non-STEM Majors Copyright © by Amy Collins Montalbano. All Rights Reserved.

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