math 1111 { lecture notes 1 equations and inequalities 1111 { lecture notes 1 equations and...

MATH 1111 – Lecture Notes

1 Equations and Inequalities

1.1 Basic Equations

Definition 1.1 A linear equation in one variable is an equation that isequivalent to one of the form

ax + b = 0

where a and b are real numbers and x is a variable.

1. Solve the equation 3x + 2 = 11.

2. Solve the equation 5x− 1 = 3x + 17.

1

3. Solve the equation 2(1− x) = 3(1 + 2x) + 5.

4. Solve the equation2

t + 6=

3

t− 1.


3− t+

4

3 + t+

16

9− t2= 0.

2

Rule 1.2 The power equation xn = a has the solution

x = n√a if n is odd

x = ± n√a if n is even and a ≥ 0

If n is even and a < 0, the equation has no real solution.

6. Find all real solutions of the equation x2 − 24 = 0.

7. Find all real solutions of the equation (x + 2)2 = 4.

3

8. Find all real solutions of the equation 64x6 = 27.

9. Find all real solutions of the equation 3(x− 3)3 = 375.

10. Find all real solutions of the equation x4/3 − 16 = 0.

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1.2 Modeling with Equations

Guidelines for modeling with equations

• Identify the variable

• Translate from words to algebra

• Set up the model

• Solve the equation and check

1. Express the given quantity in terms of the indicated variable. Theaverage of four quiz scores if each of the first three scores is 8; q =fourth quiz score.

2. Express the given quantity in terms of the indicated variable. The sumof three consecutive integers; n = the first integer of the three.

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3. A car rental company charges $30 a day and 15 cents a mile for rentinga car. Helen rents a car for two days, and her bill comes to $108. Howmany miles did she drive?

4. A phone company charges $50 a month and 2 cents a minute for eachminute used over 400 minutes. Angela’s cell phone bill is $54.92. Howmany minutes did she use?

6

Rule 1.3 The formula for simple interest is

I = Prt

where I is the total interest earned and P is the principal deposited for t yearsat interest rate r.

5. Express the given quantity in terms of the indicated variable. Theinterest obtained after one year on an investment at 3% simple interestper year; x = number of dollars invested.

6. Angie invested $12,000, a portion earning a simple interest rate of 412%

per year and the rest earning a rate of 4% per year. After 1 year thetotal interest earned on these investments was $525. How much did sheinvest at each rate?

7

Rule 1.4 Distance = rate × time

7. Express the given quantity in terms of the indicated variable. The ratein miles per hour it takes to travel a given distance in 4 hours; d =givendistance.

8. Bill left his house at 2:00PM and rode his bicycle down Main Streetat a speed of 12 mi/h. When his friend Mary arrived at his house at2:10PM, Bill’s mother told her the direction in which Bill had gone,and Mary cycled after him at a speed of 16 mi/h. At what time didMary catch up with Bill?

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9. A square plot of land has a building 60 ft long and 40 ft wide at onecorner. The rest of the land outside the building forms a parking lot.If the parking lot has area 12,000 ft2, what are the dimensions of theentire plot of land?

10. Bill earns $10 an hour at his job, but if he works more than 40 hoursin a week, he is paid 11

2times his regular salary for the overtime hours.

One week he made $475. How many overtime hours did he work thatweek?

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11. A movie star, unwilling to give his age, posed the following riddle toa gossip columnist: ”Seven years ago, I was eleven times as old as mydaughter. Now I am four times as old as she is.” How old is the moviestar?

1.3 Quadratic Equations

Definition 1.5 A quadratic equation is an equation of the form

ax2 + bx + c = 0

where a, b, and c are real numbers with a 6= 0.

Rule 1.6 Zero-Product Property

xy = 0 if and only if

10

1. Solve the quadratic equation x2 − 7x + 12 = 0 by factoring.

2. Solve the quadratic equation 3x2 + 1 = 4x by factoring.

Rule 1.7 Completing the Square To make x2 + bx a perfect square, add(b

2

)2

to both sides of your equation. This gives

x2 + bx +

(b

2

)2

=

11

3. Solve the quadratic equation x2−4x+2 = 0 by completing the square.

4. Solve the quadratic equation 2x2 + 8x = −1 by completing the square.

12

5. Solve the quadratic equation x2 − 7x = 5 by completing the square.

6. Solve the quadratic equation 5x2 − 10x + 25 = 0 by completing thesquare.

13

Rule 1.8 Quadratic Formula The solutions of the quadratic equationax2 + bx + c = 0, where a 6= 0 are

x =−b±

√b2 − 4ac

2a

7. Solve the quadratic equation 2x2 − 3 = x by using the quadratic for-mula.

8. Solve the quadratic equation x2 + 3x + 1 = 0 by using the quadraticformula.

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9. Solve the quadratic equation x2 − 2x− 15 = 0.

10. Solve the quadratic equation z2 − 32z + 9

16= 0.

11. Solve the quadratic equation 3y2 + y = 2

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Rule 1.9 The discriminant of the the equation ax2 + bx + c = 0 is

D =

If D > 0, the equation has two real solutions. If D < 0, the equation has noreal solution. If D = 0, the equation has exactly one real solution.

12. Use the discriminant to determine the number of real solutions of theequation x2 = 8x + 5. Do not solve the equation.

13. Use the discriminant to determine the number of real solutions of theequation 4x2 + 5x + 25

2= 0. Do not solve the equation.

14. Use the discriminant to determine the number of real solutions of theequation 9x2 − 6x = −1. Do not solve the equation.

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1.5 Other Types of Equations

• Polynomial Equations

• Equations Involving Radicals

• Equations of Quadratic Type

• Applications

1. Find all real solutions of the equation x3 = 16x.

2. Find all real solutions of the equation x5 + 8x2 = 0.

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3. Find all real solutions of the equation (x + 1)5 − 9(x + 1)3 = 0.

4. Find all real solutions of the equation x3 − 5x2 − 2x + 10 = 0.

5. Find all real solutions of the equation1

x− 1+

1

x + 2=

5

4.

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6. Find all real solutions of the equation√

2x + 1 + 1 = x.

Definition 1.10 An equation of quadratic type has the form

aW 2 + bW + c = 0

where a, b, and c are real numbers with a 6= 0.

7. Find all real solutions of the equation x4 − 13x2 + 40 = 0.

19

8. Find all real solutions of the equation x6 − 2x3 − 3 = 0.

9. Find all real solutions of the equation x− 4√x− 1 = 0.

20

10. A group of friends decides to buy a vacation home for $120,000, sharingthe cost equally. If they can find one more person to join them, eachperson’s contributions will drop by $6,000. How many people are inthe group?

11. A box has volume of 6 ft3. Its length is 1 ft less than its height, and itswidth is 3 ft greater than its height. What are the dimensions of thebox?

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1.6 Inequalities

Rule 1.11 An inequality looks like an equation where the = is replacedwith <, >, ≤, or ≥. The solution to an inequality is the set of values thatmake the inequality true. The solutions looks like an inequality or set ofinequalities with the variable isolated on one side.

1. Let S = {−2,−1, 0, 12, 1,√

2, 2, 4}. Determine which elements of Ssatisfy the inequality 1 < 2x− 4 ≤ 7.

2. Let S = {−2,−1, 0, 12, 1,√

2, 2, 4}. Determine which elements of Ssatisfy the inequality x2 + 2 < 4.

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Rule 1.12 • Addition and Subtraction Adding or subtraction a givenquantity on each side of an inequality gives an equivalent inequality.

• Multiplication and Division with a POSITIVE value Multi-plying or dividing by a positive quantity on each side of an inequalitygives an equivalent inequality.

• Multiplication and Division with a NEGATIVE value Multi-plying or dividing by a negative quantity on each side of an inequalityreverses the direction of the inequality.

3. Solve the linear inequality 2x − 5 > 3. Express the solution usinginterval notation and graph the solution set.

4. Solve the linear inequality 3x+11 ≤ 6x+8. Express the solution usinginterval notation and graph the solution set.

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5. Solve the linear inequality 4 − 3x ≤ −1(1 + 8x). Express the solutionusing interval notation and graph the solution set.

6. Solve the linear inequality −1 < 2x−5 < 7. Express the solution usinginterval notation and graph the solution set.

7. Solve the linear inequality 23≥ 2x−3

12≥ 1

6. Express the solution using

interval notation and graph the solution set.

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Guidelines for solving nonlinear inequalities

• Move all terms to one side

• Factor

• Find the intervals

• Make a table or diagram

• Solve

8. Solve the nonlinear inequality (x− 5)(x+ 4) ≥ 0. Express the solutionusing interval notation and graph the solution set.

9. Solve the nonlinear inequality x2 − 3x − 18 ≤ 0. Express the solutionusing interval notation and graph the solution set.

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10. Solve the nonlinear inequality 3x2−3x < 2x2 +4. Express the solutionusing interval notation and graph the solution set.

11. Solve the nonlinear inequality x3−16x < 0. Express the solution usinginterval notation and graph the solution set.

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12. Solve the nonlinear inequality (2x − 1)(−x + 3) ≥ 0. Express thesolution using interval notation and graph the solution set.

1.7 Absolute Value Equations and Inequalities

Rule 1.13

|x| = C is equivalent to

1. Solve the equation |4x| = 24.

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2. Solve the equation 5|x|+ 3 = 28.

3. Solve the equation |x + 4| = 12.

4. Solve the equation 8 + 5|13x− 5

6| = 33.

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5. Solve the equation |x− 1| = |3x + 2|.

Rule 1.14 Properties of Absolute Value Inequalities

Inequality Equivalent Form|x| < c −c < x < c|x| ≤ c −c ≤ x ≤ c|x| > c x < −c and c < x|x| ≥ c x ≤ −c and c ≤ x

6. Solve the inequality |x − 5| ≤ 3. Express the answer using intervalnotation.

29

7. Solve the inequality |x+12| ≥ 4. Express the answer using interval nota-

tion.

8. Solve the inequality 2|12x + 3| + 3 ≤ 51. Express the answer using

interval notation.

30

9. Solve the inequality 0 < |x− 5| < 2. Express the answer using intervalnotation.

10. Solve the inequality 0 < |2x − 6| < 10. Express the answer usinginterval notation.

31

11. Solve the inequality | 1x+7| > 2. Express the answer using interval nota-

tion.

12. Solve the inequality | 42x−5 | ≥ 1. Express the answer using interval

notation.

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13. Solve the inequality | 1x−9 | < 1. Express the answer using interval nota-

tion.

2 Coordinates and Graphs

2.1 The Coordinate Plane

Vocabulary

• Cartesian plane

• x-axis

• y-axis

• quadrants

• origin

• ordered pair

• x-coordinate

• y-coordinate

33

1. Plot the points (2, 3), (−2, 3), (4, 5), (4,−5), (−4, 5), (−4,−5) in the co-ordinate plane.

2. Sketch the region given by the set {(x, y)|x = 3}.

3. Sketch the region given by the set {(x, y)|1 < y < 2}.

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4. Sketch the region given by the set {(x, y)|x ≥ 1 and y < 3}.

5. Sketch the region given by the set {(x, y)|x < −2 and y < 5}.

Theorem 2.1 The Distance Formula The distance between the pointsA(x1, y1) and B(x2, y2) is

D(A,B) =

35

Theorem 2.2 The Midpoint Formula The midpoint of the line segmentthat joins the points A(x1, y1) and B(x2, y2) is(

x1 + x2

2,y1 + y2

2

)

6. Plot the points (−3,−6), (4, 18) in the coordinate plane. Find thedistance between them. Find the midpoint of the line segment thatjoints them.

7. Plot the points (0, 8), (6, 16) in the coordinate plane. Find the distancebetween them. Find the midpoint of the line segment that joints them.

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8. Plot the points (5, 0), (0, 6) in the coordinate plane. Find the distancebetween them. Find the midpoint of the line segment that joints them.

9. Plot the points (6,−2), (−1, 3) in the coordinate plane. Find the dis-tance between them. Find the midpoint of the line segment that jointsthem.

37

10. Which of the points (4, 9), (−7, 6) is closer to the origin?

11. Which of the points (−6, 3), (3, 0) is closer to the point (−2, 1)?

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12. Graph the triangle with vertices (6,−7), (11,−3), (2,−2). Find its area.What else can you say about this triangle?

2.2 Graphs of Equations in Two Variables

Definition 2.3 The graph of an equation in variables x and y is the set ofall points in the Cartesian plane (x, y) that satisfy the equation.

1. Determine whether the points (0, 0), (0, 1), (−1,−1) are on the graphof the equation x− 2y − 1 = 0.

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Definition 2.4 The x-intercepts are the x-coordinates of the points wherethe graph of an equation intersects the x-axis. The y-intercepts are the y-coordinates of the points where the graph of an equation intersects the y-axis.

How to find intercepts

2. Make a table of values and sketch the graph of the equation 2x−y = 6.Find the x- and y- intercepts.

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3. Make a table of values and sketch the graph of the equation y = x2−9.Find the x- and y- intercepts.

4. Make a table of values and sketch the graph of the equation x2+y2 = 25.Find the x- and y- intercepts.

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5. Find the x- and y-intercepts of the equation x2 − xy + y = 1.

6. Find the x- and y-intercepts of the equation xy = 5.

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Definition 2.5 The standard form of the equation of a circle with center(h, k) and radius r is

(x− h)2 + (y − k)2 = r2.

7. Find the center and radius of the circle (x− 3)2 + (y − 4)2 = 49.

8. Find the center and radius of the circle x2 + (y + 6)2 = 81.

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9. Find the equation of the circle with center (1,−2) containing the point(1, 1).

10. Find the equation of the circle that satisfies the condition that theendpoints of a diameter are P (−1, 1) and Q(5, 9).

44

11. Show the equation x2 + y2− 4x+ 10y + 13 = 0 represents a circle, andfind the center and radius of the circle.

12. Show the equation x2 + y2 + x = 0 represents a circle, and find thecenter and radius of the circle.

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Definition 2.6 A graph is symmetric about the x-axis if the graph isunchanged when reflected about the x-axis. A graph is symmetric aboutthe y-axis if the graph is unchanged when reflected about the y-axis. Agraph is symmetric about the origin if the graph is unchanged whenrotated 180◦ about the origin.

How to test for symmetry

46

13. Test the equation for symmetry.

(a) y = x4 + x2

(b) y = x3 + 10x

(c) y = x2 + 1

(d) x + y4 − y2

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2.4 Lines

Definition 2.7 The slope of a line that passes through the points A(x1, y1)and B(x2, y2) is

m =

1. Find the slope of the line through P (1,−3) and Q(−1, 6).

2. Find the slope of the line through P (−1,−4) and Q(5, 0).

Definition 2.8 The Point-Slope Equation of a Line that passes throughthe point (x1, y1) and has slope m is

y − y1 = m(x− x1).

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The Slope-Intercept Equation of a Line that has slope m and y-interceptb is

y =

The Equation of a Vertical Line through the point (a, b) is

x =

The Equation of a Horizontal Line through the point (a, b) is

y =

Definition 2.9 Two lines with slopes m1 and m2 are parallel if

m1 =

Two lines with slopes m1 and m2 are perpendicular if

m1 =

Also a horizontal line is perpendicular to a vertical line.

3. Find an equation of the line that satisfies the given conditions. Through(−2, 4); slope −1.

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4. Find an equation of the line that satisfies the given conditions. Through(−1, 2) and (4, 3).

5. Find an equation of the line that satisfies the given conditions. x-intercept 1; y-intercept -3.

6. Find an equation of the line that satisfies the given conditions. Through(2,−1) and parallel to the line x = 5.

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7. Find an equation of the line that satisfies the given conditions. Through(4, 3); parallel to the x-axis.

8. Find an equation of the line that satisfies the given conditions. Through(12,−2

3); perpendicular to the line 4x− 8y = 1.

9. Find an equation of the line that satisfies the given conditions. x-intercept 0; parallel to the line 2x− y = 6.

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10. Find the slope and y-intercept of the line 3x − 2y = 12 and draw itsgraph.

11. Find the slope and y-intercept of the line x = 9 and draw its graph.

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3 Functions

3.1 What Is a Function?

Definition 3.1 A function f is a that assigns to each ele-ment x in a set A exactly one element, called f(x), in a set B.

1. Example

(a) Rule

(b) Machine

(c) Arrow Diagram

Vocabulary

• image of x under f

• domain

• range

• independent variable

• dependent variable

• input

• output

53

There are four ways to represent a function.

• V

• A

• G

• N

2. Express the rule, “Subtract 5, then square” in function notation.

3. Express the function g(x) = x3− 4 verbally and visually.

54

4. Express the function f(x) = 2(x− 1)2 numerically using x-values fromthe set {−1, 0, 1, 2, 3}.

5. Evaluate the piecewise defined function at the valuesf(−2), f(−1), f(0), f(1).

f(x) =

{x2 if x < 0x + 1 if x ≥ 0

6. Evaluate the piecewise defined function at the valuesf(−4), f(−3

2), f(−1), f(0), f(25).

f(x) =

x2 + 2x if x ≤ −1x if − 1 < x ≤ 1−1 if x > 1

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Definition 3.2 The domain of a function is the set of real numbers onwhich the function is .

7. Find the domain of the function g(x) = 2x + 1.

8. Find the domain of the function h(x) = 3x on 0 ≤ x < 5.

56

9. Find the domain of the function f(x) =1

x2 + x− 6.

10. Find the domain of the function h(x) =3

x + 1.

11. Find the domain of the function g(x) =√x− 5.

57

12. Find the domain of the function f(x) =1√

3− x.

13. Find the domain of the function g(x) =

√4− x

x + 7.

14. Find the domain of the function f(x) =

√2 + x

3− x.

58

3.2 Graphs of Functions

1. Sketch the graph of the function f(x) = 2x− 4 by first making a tableof values.

2. Sketch the graph of the function g(x) = 16−x2 by first making a tableof values.

59

3. Sketch the graph of the function h(x) = |x+ 1| by first making a tableof values.

4. Sketch the graph of the function g(x) = 1 +√x by first making a table

of values.

60

Basic graphs of some functions

• Linear functions

• Power functions

• Root functions

• Reciprocal functions

• Absolute value functions

61

5. Sketch the graph of the piecewise defined function.

f(x) =

{3 if x < 2x− 1 if x ≥ 2


f(x) =

{x2 if x ≤ 01 if x > 0

62


f(x) =

{2x + 3 if x < −13− x if x ≥ −1

Rule 3.3 The Vertical Line TestA curve in the coordinate plane is the graph of a function if and only if no ver-tical line the curve more than .

63

8. Use the Vertical Line Test to determine whether the curve is the graphof a function of x. If it is, give the domain and range of the function.


64


11. Does the equation define y as a function of x?

(a) y − x2 = 2

(b) y2 − x = 2

65

3.3 Getting Information from the Graph of a Function

1. The graph of a function f is given below.

(a) Find f(−1), f(1), f(4).

(b) Find the domain and range of f .

(c) For which values of x is f(x) = 0?

2. The graph of a function f is given below.

(a) Find f(0), f(2), f(−2).

(b) Find the domain and range of f .

(c) For which values of x is f(x) = 3?

66

3. Graphs of the functions f and g are given below.

(a) Which is larger g(1) or f(1)?

(b) For which values of x is f(x) = g(x)?

Definition 3.4 • f is increasing on an interval I ifwhenever in I.

• f is decreasing on an interval I if wheneverin I.

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4. Determine the intervals on which the function given below is increasingand decreasing.

5. Determine the intervals on which the function given below is increasingand decreasing.

68

Definition 3.5 • The function value f(a) is a local maximum valueof f if when x is near a. In this case we say f has alocal maximum at x = a.

• The function value f(b) is a local minimum value of f ifwhen x is near b. In this case we say f has a local minimum at x = b.

6. Determine the intervals on which the function given below is increasingand decreasing. Final all local maximum and minimum values of thefunction and the value of x at which each occurs.

69

7. Determine the intervals on which the function given below is increasingand decreasing. Final all local maximum and minimum values of thefunction and the value of x at which each occurs.

8. The graph shows the speed of a car commuting to work. Determinethe intervals on which the function is increasing and decreasing. Whathappened between 7:15 and 7:35? What happened at 7:10?

70

9. Three runners compete in a 100-meter hurdle race. The graph depictsthe distance run as a function of time for each runner. What does thisgraph tell you about the race? Who won? Did each runner finish?What happened to runner B?

3.4 Average Rate of Change of a Function

Definition 3.6 The average rate of change of a function y = f(x) be-tween x = a and x = b is

average rate of change =f(b)− f(a)

b− a.

The average rate of change is the slope of the line betweenx = a and x = b.

71

1. The graph of a function is given. Determine the average rate of changeof the function between the indicated points on the graph.

2. The graph of a function is given. Determine the average rate of changeof the function between the indicated points on the graph.

72

3. Determine the average rate of change of the function f(x) = x2 + 2xbetween x = −1 and x = 4.

4. Determine the average rate of change of the function h(x) =√x be-

tween x = 4 and x = 16.

5. Determine the average rate of change of the function f(x) =2

x + 1between x = −0 and x = 3.

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6. The table shows the number of CD players sold in a small electronicsstore in the years 1993-2003.

Y ear CD players sold1993 5121994 5201995 4131996 4101997 4681998 5101999 5902000 6072001 7322002 6122003 584

(a) What was the average rate of change of sales between 1993 and2003?

(b) What was the average rate of change of sales between 1994 and1996?

(c) Between which two successive years did sales increase most quickly?Decrease most quickly?

74

3.5 Transformations of Functions

Vertical Shifts of GraphsSuppose c > 0

• To graph y = f(x) + c, shift the graph of y = f(x)c units.

• To graph y = f(x)− c, shift the graph of y = f(x)c units.

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Horizontal Shifts of GraphsSuppose c > 0

• To graph y = f(x+c), shift the graph of y = f(x) to thec units.

• To graph y = f(x−c), shift the graph of y = f(x) to thec units.

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Reflecting Graphs

• To graph y = −f(x), reflect the graph of y = f(x) in the .

• To graph y = f(−x), reflect the graph of y = f(x) in the .

77

Vertical Stretching and Shrinking of GraphsTo graph y = cf(x):

• If c > 1, the graph of y = f(x) vertically by a factorof c.

• If 0 < c < 1, the graph of y = f(x) vertically by afactor of c.

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Horizontal Stretching and Shrinking of GraphsTo graph y = f(cx):

• If c > 1, the graph of y = f(x) horizontally by a factorof 1/c.

• If 0 < c < 1, the graph of y = f(x) horizontally by afactor of 1/c.

79

1. Sketch the graph of the function f(x) = |x − 1| + 5, not by plottingpoints, but by starting with the graph of a standard function and ap-plying transformations.

2. Sketch the graph of the function f(x) =√−x + 3, not by plotting

points, but by starting with the graph of a standard function and ap-plying transformations.

80

3. Sketch the graph of the function f(x) = −(x + 1)2, not by plottingpoints, but by starting with the graph of a standard function and ap-plying transformations.

4. Sketch the graph of the function f(x) = 4x3−1, not by plotting points,but by starting with the graph of a standard function and applyingtransformations.

81

5. A function f is given, and the indicated transformations are appliedto its graph in the given order. Write and equation for the final trans-formed graph.f(x) = |x|; shift 3 units to the right and shift upward 1 unit

6. A function f is given, and the indicated transformations are appliedto its graph in the given order. Write and equation for the final trans-formed graph.f(x) = x2; stretch vertically by a factor of 2, shift downward 2 units,and shift 3 units to the right

82

7. The graphs of f and g are given. Find a formula for the function g.

8. The graphs of f and g are given. Find a formula for the function g.

83

3.6 Combining Functions

Let f and g be functions with domains A and B. Then

(f + g)(x) = Domain

(f − g)(x) = Domain

(fg)(x) = Domain

(f

g

)(x) = Domain

1. Find f+g, f−g, fg, and f/g and their domains where f(x) =√

4− x2

and g(x) =√

1− x.

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2. Find f + g, f − g, fg, and f/g and their domains where f(x) =2

x + 1

and g(x) =x

x + 1.

Definition 3.7 Given two function f and g, the composition of f and g,written f ◦ g is defined by

(f ◦ g)(x) = .

The domain of f ◦ g is all x that are defined for both g(x) and f(g(x)).

85

3. Use f(x) = 6x + 1 and g(x) = x2 − 3 to evaluate the expression.

(a) f(g(0))

(b) g(f(0))

(c) f(f(−1))

(d) g(g(2))

(e) (f ◦ g)(3)

(f) (g ◦ f)(2)

4. Use the graphs of f and g to evaluate the expresssion.

(a) f(g(0))

(b) g(f(0))

(c) f(f(−1))

(d) g(g(2))

(e) (f ◦ g)(3)

(f) (g ◦ f)(2)

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5. Find the functions f ◦ g, g ◦ f , f ◦ f , and g ◦ g and their domains wheref(x) = 6x− 5 and g(x) = x

2.

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6. Find the functions f ◦ g, g ◦ f , f ◦ f , and g ◦ g and their domains wheref(x) = x2 and g(x) =

√x− 1.

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7. Find the functions f ◦ g, g ◦ f , f ◦ f , and g ◦ g and their domains where

f(x) =x

x + 1and g(x) =

1

x.

89

8. Express the function F (x) = (x− 9)5 in the form f ◦ g.

9. Express the function F (x) =√x + 1 in the form f ◦ g.

10. Express the function F (x) =1

x + 3in the form f ◦ g.

90

3.7 One-to-One Functions and Their Inverses

Definition 3.8 A function with domain A is called a one-to-one functionif f(x1) 6= f(x2) whenever x1 6= x2.

Horizontal Line Test

1. Use the graph to determine if f is one-to-one.

91



92

4. Determine whether the functionf(x) = x4 + 5 is one-to-one.

5. Determine whether the functionf(x) =1

xis one-to-one.

93

Definition 3.9 Let f be a one-to-one function with domain A and rangeB. Then its inverse function f−1 has domain and range

and is defined by

f−1(y) = x ⇐⇒

How to find the inverse of a one-to-one function:

Algebraically

1.

2.

3.

4.

Graphically

1.

94

6. Find the inverse function of f(x) = 4x + 7.

7. Find the inverse function of g(x) = 5− 4x3.

95

8. Find the inverse function of h(x) = 1 +√

1 + x.

9. Find the inverse function of f(x) =2x + 5

x− 7.

96

10. Find the inverse function of f(x) =1− 3x

4 + 2x.

11. Find the inverse function of f(x) =4x + 5

x− 7.

97

Rule 3.10 Inverse Function PropertyIf f and g are inverses, then f(g(x)) = x and g(f(x)) = x.

12. Use the Inverse Function Property to show that f(x) = 3x + 6 and

g(x) =x− 6

3are inverses of each other.

13. Use the Inverse Function Property to show that f(x) = 3√x− 7 and

g(x) = x3 + 7 are inverses of each other.

98

14. Assume that f is a one-to-one function. If f(x) = 3x− 1, find f−1(8).

15. Assume that f is a one-to-one function. If f(x) =2 + x

1 + x, find f−1(2).

99

5 Exponential and Logarithmic Functions

5.1 Exponential Functions

Laws of Exponents (Review)

If s, t, a, and b are real numbers, with a > 0 and b < 0, then

• as · at = as+t

• (as)t = ast

• (ab)s = asbs

• (ab)s = as

bs

• a−s = 1as

= ( 1a)s

1. A frog population of 100 frogs is brought to a zoo. It is estimated thatthe frog population triples every year.

(a) Estimate the frog population after 2 years? 3 years?

(b) Find a function the models the number of frogs after t years.

(c) Estimate the frog population after 20 years.

100

Definition 5.1 An exponential function is a function of the formwhere the base a is a positive real number not equal to one. The domain off is the set of all real numbers.

2. Graph the exponential function f(x) = 2x.

3. Graph the exponential function f(x) = (12)x.

101

4. Graph the exponential functions f(x) = 2x and g(x) = 7x on the samecoordinate plane.

5. Graph the exponential functions f(x) = (12)x and g(x) = (1

5)x on the

same coordinate plane.

102

6. Find the exponential function f(x) = ax whose graph is given.

7. Find the exponential function f(x) = ax whose graph is given.

103

8. Graph f(x) = 2−x−3 and determine the domain, range, and horizontalasymptote of f .

9. Graph f(x) = (13)x+5 and determine the domain, range, and horizontal

asymptote of f .

104

Compound Interest

The amount A after t years due to a principal P invested at an annualinterest rate r compounded n times per year is

A(t) =

where A(t)=amount after t years, P=Principal, r=interest rate, n= numberof times interest is compounded per year, and t= time of years.

10. Investing $1000 at an annual rate of 12% compounded annually, semi-annually, quarterly, monthly, and daily will yield the following amountsafter 3 years.

(a) Annual compounding:

(b) Semiannual compounding:

(c) Quarterly compounding:

(d) Monthly compounding:

(e) Daily compounding:

105

5.2 The Natural Exponential Function

Definition 5.2 The number e is defined as the number that the expression(1 +

1

n

)n

approaches as n approaches infinity. In calculus, this is expressed

using limit notation as

e = limn→∞

(1 +1

n)n.

Definition 5.3 The natural exponential function is the functionwith base e. The domain of f is the set of all real numbers.

1. Graph f(x) = −ex+3 and determine the domain, range, and horizontalasymptote of f .

106

2. Graph f(x) = ex−2+4 and determine the domain, range, and horizontalasymptote of f .

3. The atmospheric pressure p on a balloon or plane decreases with in-creasing height. This pressure, measured in millimeters of mercury, isrelated to the number of kilometers h above sea level by the function

p(h) = 760e−0.345h.

(a) Find the atmospheric pressure at a height of 2 kilometers (over 1mile).

(b) What is the pressure at a height of 10 kilometers?

107

Continuous Compounding

The amount A after t years due to a principal P invested at an annualinterest rate r compounded continuously is

A(t) = Pert.

4. Compute the amount A that results from investing a principal P of$1000 at an annual rate r of 12% compounded continuously for a timet of 3 years.

5. Set the following problems up but do not solve down at this time:

(a) How long will it take for an investment to double in value if itearns 5% compounded continuously?

(b) How long will it take to triple at this rate?

5.3 Logarithmic Functions

Definition 5.4 The logarithmic function to the base a, where a > 0 anda 6= 1, is denoted by

y = loga x

(read as y is the logarithm to the base a of x) and is defined by

y = loga x ⇐⇒ x = ay.

The domain of the logarithmic function y = loga x is (0,∞).

1. Express the equation in logarithmic form.

(a) 4 = log3 81

(b) 1.23 = m

(c) eb = 9

(d) a4 = 24

108

2. Express the equation in exponential form.

(a) −1 = log5(15)

(b) loga 4 = 5

(c) loge b = −3

(d) log3 5 = c

3. Evaluate the following logarithmic expressions.

(a) log10 1000

(b) log10 0.1

(c) log16 4

(d) log2 16

(e) log3127

109

Properties of Logarithms

• loga 1 = 0

• loga a = 1

• loga ax = x

• aloga x = x

4. Evaluate the following logarithmic expressions.

(a) log3 1

(b) log5 58

(c) log2 2

(d) 4log4 12

(e) log7 7x+1

(f) 9log9(2−x)

110

Rule 5.5 The domain of the logarithmic function = the range of the expo-nential function = andthe range of the logarithmic function = the domain of the exponential func-tion = .

5. Sketch the graph of f(x) = log2 x and g(x) = log5 x.

6. Sketch the graph of f(x) = − log2 x and g(x) = log2(−x).

111

7. Find the domain and sketch the graph of the function f(x) = 2 +log5(x + 1).

8. Find the domain and sketch the graph of the functiong(x) = − log10(x− 3).

112

9. Find the function of the form y = loga x whose graph is given.

10. Find the function of the form y = loga x whose graph is given.

113

Definition 5.6 If the base of the logarithmic function is the number e, thenwe have the natural logarithmic function y = loge x = lnx.If the base of a logarithmic function is the number 10, then we have thecommon logarithm function y = log10 x = log x.

11. Use the definition of the logarithmic function to findx.

(a) log3(4x− 7) = 2

(b) logx 64 = 2

(c) log7(5x + 4) = 2

(d) log2 x = 2

(e) e2x = 5

(f) 103x = 6

114

5.4 Laws of Logarithms

In the following properties, M , N , and a are positive real numbers, witha 6= 1 , and r is any real number.

• The Log of a Product Equals the Sum of the Logs

loga(MN) =

• The Log of a Quotient Equals the Difference of the Logs

loga

(M

N

)=

• The Log of a Power Equals the Product of the Power and the Log

logaMr =

1. Evaluate each expression.

(a) log4 2 + log4 32

(b) log2 80− log2 5

115

(c) −13

log2 8

(d) log3 951

(e) log3

√27

2. Use the Laws of Logarithms to expand each expression.

(a) log2(6x)

(b) ln(x(x− 1))

116

(c) log5(x3y6)

(d) ln(

ab3√c

)

(e) log(

x3√x−1

3x−4

)

3. Use the Laws of Logarithms to combine each expression.

(a) log3 5 + 5 log3 2

(b) 3 log x + 12 log(x + 1)

117

(c) ln(x + y) + ln(x− y)− 4 ln z

(d) 3 ln s + 12

ln t− 4ln(t2 + 1)

(e) 13

log(x + 2)3 + 12[log x4 − log(x2 − x− 6)2]

Change of Base Formula

logb x =loga x

loga b

118

4. Use the Change of Base Formula and common or natural logarithmsto evaluate each logarithm, correct to five decimal places.

(a) log8 5

(b) log9 20

5.5 Exponential and Logarithmic Equations

Guidelines for solving exponential equations:

• Isolate the exponential expression on one side of the equation.

• Take the logarithm of each side.

• Solve for the variable.

1. Solve the equation 5−2x = 7.

119

2. Solve the equation 3x+4 = 6 .

3. Solve the equation 8e9x = 20.

4. Solve the equation 93−2x = 4.

120

5. Solve the equation 3xex + x2ex = 0.


1 + e−x= 4.

121

Guidelines for Solving Logarithmic Equations:

• Isolate the logarithmic term on one side of the equation; you might firstneed to combine the logarithmic terms.

• Write the equation in exponential form (or raise the base to each sideof the equation).

• Solve for the variable.

7. Solve the equation lnx = 8.

8. Solve the equation log2(25− x) = 3.

122

9. Solve the equation 4 + 3 log(2x) = 16.

10. Solve the equation log2(x2 − x− 2) = 2.

11. Solve the equation log(x + 2) + log(x− 1) = 1.

123

12. Solve the equation log2 x + log2(x− 3) = 2.

13. Find the inverse function of f when f(x) = 23x.

14. Find the inverse function of f when f(x) = log3(x− 1).

124

15. A man invests $2,000 at an interest rate of 6% per year compoundedcontinuously.

(a) What is the amount after 2 years?

(b) How long will it take for the amount to be $10,000?

16. A sum of $5,000 is invested at an interest rate of 5% per year. Find thetime required for the money to double if the interest is compoundedaccording to the following method.

(a) Semiannual

(b) Continuous

125

17. A sum of $1,000 is invested at an interest rate of 4% per year. Findthe time required for the amount to grow to $4,000 if interest is com-pounded continuously.

5.6 Modeling with Exponential and Logarithmic Equa-tions

Exponential Growth Model

n(t) = n0ert

where n(t) is the population at time t, n0 is the initial size of the population,and r is the relative rate of growth.

126

1. A bacteria initially has 100 bacteria and is observed to double every1.5 hours.

(a) Find an exponential function n(t) for the number of bacteria aftert hours.

(b) How many bacteria are there after 5 hours?

(c) When will the bacteria count reach 20,000?

127

2. The population of certain species of deer triples every 8 years. This typeof deer was introduced into Georgia 24 years and now the populationis 135.

(a) What was the initial size of the deer population?

(b) What will the population be 3 years from now?

(c) Sketch a graph of the deer population.

128

3. The population of a city has a relative growth rate of 4% per year. Thecity council is trying to reduce the growth rate to 3.5%. The populationin 2011 is 135,000. Find the projected population for the year 2021 forthe following conditions.

(a) The relative growth rate remains a 4% per year.

(b) The relative growth rate is decreased to 3.5% per year.

4. The population of the United States was 312,448,000 in 2011 and therate of growth was 0.85% per year.

(a) By what year will the population have doubled?

(b) By what year will the population have tripled?

129

5. The graph shows the mosquito population for the next few years. As-sume that the population grows exponentially.

(a) What was the initial population?

(b) Find a function that models the population in terms of t yearsfrom now.

(c) What is the projected population after 10 years?

130

Radioactive Decay Model

m(t) = m0e−rt

where m(t) is the mass at time t, m0 is the initial mass, and r is the rate ofdecay.

6. The half-life of Flourine-21 is 4 seconds. Suppose we have a 20 mgsample.

(a) Find a function m(t) that models the mass remaining after t sec-onds.

(b) How much of the sample will remain after 7 seconds?

(c) After how long will only 1 mg of the sample remain?

131

7. If a 400 g sample of a radioactive element decays to 300 g in 6 hours,find the half-life of the element.

8. A human fossil has 65% of the carbon-14 that would be present in aliving human being. If the half-life of carbon-14 is 5,730 years, how oldis the fossil?

132

Newton’s Law of Cooling

T (t) = Ts + D0e−kt

where T (t) is the temperature at time t, D0 is the initial difference in tem-perature of an object and it’s surrounding temperature Ts, and k is positiveconstant.

9. A turkey is taken from an oven when it’s temperature is 185◦ F and isplaced in a room where the temperature is 70◦ F.

(a) If the temperature of the turkey is 150◦ F after half an hour, whatis it’s temperature after 45 minutes?

(b) When will the turkey cool to 100◦ F?

133

10. Normal body temperature is 98.6◦F. Immediately after death the bodystarts to cool. It has been determined that the constant in Newton’sLaw of Cooling is k = 0.1947 with time measured in hours. Supposethe surrounding temperature is 60◦F.

(a) Find a function T (t) that models temperature t hours after death.

(b) If the temperature of the body is now 72◦F, how long ago wastime of death?

11 Systems of Equations and Inequalities

11.1 Systems of Linear Equations in Two Variables

Definition 11.1 A system of equations is a collection of two or moreequations, each containing one or more variables. A solution to a systemof equations consists of values for the variables that are solutions of eachequation of the system.

134

Steps for Solving by Substitution

• Pick one of the equations and solve for one of the variables in terms ofthe remaining variables.

• Substitute into the OTHER equation.

• Solve this equation.

• Back-substitute.

1. Solve {2x + y = 1

3x + 4y = 14

2. Solve {x2 + y2 = 1003x− y = 10

135

Elimination Method

• Adjust the coefficients by multiplying one or more of the equationsby appropriate numbers so that the coefficient of one variable in oneequation is the negative of its coefficient in the other equation.

• Add the equations to eliminate one variable, then solve for the remain-ing variable.

• Back-substitute and and solve for the remaining variable.

3. Solve {3x + 2y = 14x− 2y = 2

4. Solve {x2 + 3y = 103x + 2y = 10

136

Number of solutions of a linear system in two variables

• The system has exactly one solution.

• The system has no solution.

• The system has infinitely many solutions.

5. Graph the linear system to determine if it has one solution, no solution,or an infinite number of solutions. If there is exactly one solution, usethe graph to find it. {

x + 2y = 42x + 4y = 8

137

6. Graph the linear system to determine if it has one solution, no solution,or an infinite number of solutions. If there is exactly one solution, usethe graph to find it. {

2x− y = 03x + 2y = 7

7. Solve the system or show that is has no solution. If the system hasinfinitely many solutions, express them in ordered pair form.{

4x + 5y = 1010x + y = 11

138


2x + 4y = 1−5x− 10y = 3


3x− 5y = 35x + 5y = 21

139


7x− y = 4−21x + 3y = −12

13 Sequences and Series

13.2 Arithmetic Sequences

Definition 13.1 A sequence is a function f whose domain is the set ofnatural numbers. The values f(1), f(2), f(3), . . . are called the terms of thesequence.

1. Find the first five terms and the 100th term of the sequence defined byeach formula.

(a) an = 2n− 1

140

(b) cn = n2 − 1

(c) rn =(−1)n

2n

Definition 13.2 An arithmetic sequence is a sequence of the form

a, a + d, a + 2d, a + 3d, . . .

The number a is the first term and d is the common difference of thesequence. The nth term of an arithmetic sequence is given by

an = a + (n− 1)d

141

2. Find the first six terms, the common difference d, the nth term, andthe 300th term of the arithmetic sequence.

(a) 2, 5, . . .

(b) 13, 7, . . .

(c) Given by a = 1 and d = 4

142

Definition 13.3 For the sequence

a1, a2, a3, . . . , an, . . .

the partial sums are

S1 = a1

S2 = a1 + a2

S3 = a1 + a2 + a3...

Sn = a1 + a2 + a3 + · · ·+ an...

S1 is called the first partial sum, S2 is the second partial sum, and soon. Sn is called the nth partial sum. The sequence S1, S2, S3, . . . is calledthe sequence of partial sums.

3. Find S1, S2, and S3 for the sequences in Problem 1.

143

Rule 13.4 Partials Sums of an Arithmetic Sequence are given by the follow-ing formulas.

Sn =n

2(2a + (n− 1)d) or

Sn = n

(a + an

2

)

4. Find S3 and Sn for the sequences in Problem 2.

144

5. Find the sum.

(a) 3 + 8 + 13 + · · ·+ 278

(b) −10− 9.9− 9.8− · · · − 0.1

13.3 Geomentric Sequences

Definition: A geometric sequence is a sequence of the form

a, ar, ar2, ar3, . . .

The number a is the first term, and r is the common ratio of the sequence.The nth term of a geometric sequence is given by

an = arn−1

145

1. Find the first five terms, the common ration r, the nth term, and the250th term of the geometric sequence.

(a) 5, 15, 45, . . .

(b) 1, 13, 19, . . .

(c) Given by a = 2 and r = −5

146

Rule 13.5 Partial Sums of a Geometric Sequence are given by the formula

Sn = a1− rn

1− r.

2. Find S3 and Sn for the sequences in Example 1.

147

3. Find the sum.

(a) 2 + 8 + 32 + 128 + · · ·+ 2, 097, 152

(b) 1− 12

+ 14− 1

8+ · · · − 1

512

148

4. Determine if the following sequences are arithmetic, geometric, or nei-ther. Find the next term of the sequence.

(a)√

2, 2√

2, 3√

2, 4√

2, . . .

(b) 1,−32, 2,−5

2, . . .

(c) 3, 1, 13, 19, . . .

(d) 2, 2.75, 3.5, 4.25, . . .

(e) 34, 12, 13, 29, . . .

149

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