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College Algebra

Review Exercises

College AlgebraReview Exercises
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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 7.1 Systems of Linear Equations: Two Variables
    3. 7.2 Systems of Linear Equations: Three Variables
    4. 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 7.4 Partial Fractions
    6. 7.5 Matrices and Matrix Operations
    7. 7.6 Solving Systems with Gaussian Elimination
    8. 7.7 Solving Systems with Inverses
    9. 7.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  9. 8 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 8.1 The Ellipse
    3. 8.2 The Hyperbola
    4. 8.3 The Parabola
    5. 8.4 Rotation of Axes
    6. 8.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  10. 9 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 9.1 Sequences and Their Notations
    3. 9.2 Arithmetic Sequences
    4. 9.3 Geometric Sequences
    5. 9.4 Series and Their Notations
    6. 9.5 Counting Principles
    7. 9.6 Binomial Theorem
    8. 9.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
  12. Index

Exponential Functions

1.

Determine whether the function y=156 ( 0.825 ) t y=156 ( 0.825 ) t represents exponential growth, exponential decay, or neither. Explain

2.

The population of a herd of deer is represented by the function A(t)=205 (1.13) t , A(t)=205 (1.13) t , where t t is given in years. To the nearest whole number, what will the herd population be after 6 6 years?

3.

Find an exponential equation that passes through the points (2, 2.25) (2, 2.25) and (5,60.75). (5,60.75).

4.

Determine whether Table 1 could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

x 1 2 3 4
f(x) 3 0.9 0.27 0.081
Table 1
5.

A retirement account is opened with an initial deposit of $8,500 and earns 8.12% 8.12% interest compounded monthly. What will the account be worth in 20 20 years?

6.

Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with 7.5% 7.5% APR, compounded daily, in order to reach her goal in 3 3 years?

7.

Does the equation y=2.294 e 0.654t y=2.294 e 0.654t represent continuous growth, continuous decay, or neither? Explain.

8.

Suppose an investment account is opened with an initial deposit of $10,500 $10,500 earning 6.25% 6.25% interest, compounded continuously. How much will the account be worth after 25 25 years?

Graphs of Exponential Functions

9.

Graph the function f(x)=3.5 ( 2 ) x . f(x)=3.5 ( 2 ) x . State the domain and range and give the y-intercept.

10.

Graph the function f(x)=4 ( 1 8 ) x f(x)=4 ( 1 8 ) x and its reflection about the y-axis on the same axes, and give the y-intercept.

11.

The graph of f(x)= 6.5 x f(x)= 6.5 x is reflected about the y-axis and stretched vertically by a factor of 7. 7. What is the equation of the new function, g(x)? g(x)? State its y-intercept, domain, and range.

12.

The graph below shows transformations of the graph of f(x)= 2 x . f(x)= 2 x . What is the equation for the transformation?

Graph of f(x)=2^x
Figure 1

Logarithmic Functions

13.

Rewrite log 17 ( 4913 )=x log 17 ( 4913 )=x as an equivalent exponential equation.

14.

Rewrite ln( s )=t ln( s )=tas an equivalent exponential equation.

15.

Rewrite a 2 5 =b a 2 5 =b as an equivalent logarithmic equation.

16.

Rewrite e 3.5 =h e 3.5 =h as an equivalent logarithmic equation.

17.

Solve for x if log 64 (x)= 1 3 log 64 (x)= 1 3 by converting to exponential form.

18.

Evaluate log 5 ( 1 125 ) log 5 ( 1 125 )without using a calculator.

19.

Evaluate log( 0.000001 ) log( 0.000001 )without using a calculator.

20.

Evaluate log(4.005) log(4.005)using a calculator. Round to the nearest thousandth.

21.

Evaluate ln( e 0.8648 ) ln( e 0.8648 )without using a calculator.

22.

Evaluate ln( 18 3 ) ln( 18 3 )using a calculator. Round to the nearest thousandth.

Graphs of Logarithmic Functions

23.

Graph the function g(x)=log( 7x+21 )4. g(x)=log( 7x+21 )4.

24.

Graph the function h(x)=2ln( 93x )+1. h(x)=2ln( 93x )+1.

25.

State the domain, vertical asymptote, and end behavior of the function g(x)=ln( 4x+20 )17. g(x)=ln( 4x+20 )17.

Logarithmic Properties

26.

Rewrite ln( 7r11st ) ln( 7r11st ) in expanded form.

27.

Rewrite log 8 ( x )+ log 8 ( 5 )+ log 8 ( y )+ log 8 ( 13 ) log 8 ( x )+ log 8 ( 5 )+ log 8 ( y )+ log 8 ( 13 )in compact form.

28.

Rewrite log m ( 67 83 ) log m ( 67 83 ) in expanded form.

29.

Rewrite ln( z )ln( x )ln( y ) ln( z )ln( x )ln( y ) in compact form.

30.

Rewrite ln( 1 x 5 ) ln( 1 x 5 ) as a product.

31.

Rewrite log y ( 1 12 ) log y ( 1 12 )as a single logarithm.

32.

Use properties of logarithms to expand log( r 2 s 11 t 14 ). log( r 2 s 11 t 14 ).

33.

Use properties of logarithms to expand ln( 2b b+1 b1 ). ln( 2b b+1 b1 ).

34.

Condense the expression 5ln( b )+ln( c )+ ln( 4a ) 2 5ln( b )+ln( c )+ ln( 4a ) 2 to a single logarithm.

35.

Condense the expression 3 log 7 v+6 log 7 w log 7 u 3 3 log 7 v+6 log 7 w log 7 u 3 to a single logarithm.

36.

Rewrite log 3 ( 12.75 ) log 3 ( 12.75 ) to base e. e.

37.

Rewrite 5 12x17 =125 5 12x17 =125 as a logarithm. Then apply the change of base formula to solve for x x using the common log. Round to the nearest thousandth.

Exponential and Logarithmic Equations

38.

Solve 216 3x 216 x = 36 3x+2 216 3x 216 x = 36 3x+2 by rewriting each side with a common base.

39.

Solve 125 ( 1 625 ) x3 = 5 3 125 ( 1 625 ) x3 = 5 3 by rewriting each side with a common base.

40.

Use logarithms to find the exact solution for 7 17 9x 7=49. 7 17 9x 7=49. If there is no solution, write no solution.

41.

Use logarithms to find the exact solution for 3 e 6n2 +1=60. 3 e 6n2 +1=60. If there is no solution, write no solution.

42.

Find the exact solution for 5 e 3x 4=6 5 e 3x 4=6 . If there is no solution, write no solution.

43.

Find the exact solution for 2 e 5x2 9=56. 2 e 5x2 9=56. If there is no solution, write no solution.

44.

Find the exact solution for 5 2x3 = 7 x+1 . 5 2x3 = 7 x+1 . If there is no solution, write no solution.

45.

Find the exact solution for e 2x e x 110=0. e 2x e x 110=0. If there is no solution, write no solution.

46.

Use the definition of a logarithm to solve. 5 log 7 ( 10n )=5. 5 log 7 ( 10n )=5.

47.

47. Use the definition of a logarithm to find the exact solution for 9+6ln( a+3 )=33. 9+6ln( a+3 )=33.

48.

Use the one-to-one property of logarithms to find an exact solution for log 8 ( 7 )+ log 8 ( 4x )= log 8 ( 5 ). log 8 ( 7 )+ log 8 ( 4x )= log 8 ( 5 ). If there is no solution, write no solution.

49.

Use the one-to-one property of logarithms to find an exact solution for ln( 5 )+ln( 5 x 2 5 )=ln( 56 ). ln( 5 )+ln( 5 x 2 5 )=ln( 56 ). If there is no solution, write no solution.

50.

The formula for measuring sound intensity in decibels D D is defined by the equation D=10log( I I 0 ), D=10log( I I 0 ), where I I is the intensity of the sound in watts per square meter and I 0 = 10 12 I 0 = 10 12 is the lowest level of sound that the average person can hear. How many decibels are emitted from a large orchestra with a sound intensity of 6.3 10 3 6.3 10 3 watts per square meter?

51.

The population of a city is modeled by the equation P(t)=256,114 e 0.25t P(t)=256,114 e 0.25t where t t is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

52.

Find the inverse function f 1 f 1 for the exponential function f( x )=2 e x+1 5. f( x )=2 e x+1 5.

53.

Find the inverse function f 1 f 1 for the logarithmic function f( x )=0.25 log 2 ( x 3 +1 ). f( x )=0.25 log 2 ( x 3 +1 ).

Exponential and Logarithmic Models

For the following exercises, use this scenario: A doctor prescribes 300 300 milligrams of a therapeutic drug that decays by about 17% 17% each hour.

54.

To the nearest minute, what is the half-life of the drug?

55.

Write an exponential model representing the amount of the drug remaining in the patient’s system after t t hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 24 24 hours. Round to the nearest hundredth of a gram.

For the following exercises, use this scenario: A soup with an internal temperature of 350° 350° Fahrenheit was taken off the stove to cool in a 71°F 71°F room. After fifteen minutes, the internal temperature of the soup was 175°F. 175°F.

56.

Use Newton’s Law of Cooling to write a formula that models this situation.

57.

How many minutes will it take the soup to cool to 85°F? 85°F?

For the following exercises, use this scenario: The equation N( t )= 1200 1+199 e 0.625t N( t )= 1200 1+199 e 0.625t models the number of people in a school who have heard a rumor after t t days.

58.

How many people started the rumor?

59.

To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?

60.

What is the carrying capacity?

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

61.
xf(x)
13.05
24.42
36.4
49.28
513.46
619.52
728.3
841.04
959.5
1086.28
62.
xf(x)
0.518.05
117
315.33
514.55
714.04
1013.5
1213.22
1313.1
1512.88
1712.69
2012.45
63.

Find a formula for an exponential equation that goes through the points ( 2,100 ) ( 2,100 ) and ( 0,4 ). ( 0,4 ). Then express the formula as an equivalent equation with base e.

Fitting Exponential Models to Data

64.

What is the carrying capacity for a population modeled by the logistic equation P(t)= 250,000 1+499 e 0.45t ? P(t)= 250,000 1+499 e 0.45t ?What is the initial population for the model?

65.

The population of a culture of bacteria is modeled by the logistic equation P(t)= 14,250 1+29 e 0.62t , P(t)= 14,250 1+29 e 0.62t , where t t is in days. To the nearest tenth, how many days will it take the culture to reach 75% 75% of its carrying capacity?

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

66.
xf(x)
1409.4
2260.7
3170.4
4110.6
574
644.7
732.4
819.5
912.7
108.1
67.
xf(x)
0.1536.21
0.2528.88
0.524.39
0.7518.28
116.5
1.512.99
29.91
2.258.57
2.757.23
35.99
3.54.81
68.
xf(x)
09
222.6
444.2
562.1
796.9
8113.4
10133.4
11137.6
15148.4
17149.3
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