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Faculty of Science

Unit 2: Functions and Graphs

MATH 1001 Pre-Calculus Mathematics

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Select questions associated with the following course textbook are used with

permission from Cengage Learning:

Swokowski, E. W., & Cole, J. A. (2012). Precalculus: Functions and graphs

(12th ed.). Belmont, CA: Brooks/Cole, Cengage Learning

Course Revision Team (2019) Course Reviser: Saeed Rahmati, PhD

Course Editor: Courtney Charlton, MA

Associate Dean, Science: Dennis Acreman, PhD

Course Development Team Course Writer: Alan Cooper, PhD

Consultant: Bernadette Harris, PhD

Course Reviser (2014): Bernadette Harris, PhD

Course Revisions/Writing (2008): Fae Debeck, MS, and Adriana Stefan, MMath

Instructional Designer: Ted Keating, MEd

Course Reference: MATH 1001_SW3

Thompson Rivers University

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Table of Contents Unit 2: Functions and Graphs

Introduction ................................................................................................................. U2‐1

Learning Outcomes ..................................................................................................... U2‐1

2.1: Rectangular Coordinate Systems ...................................................................... U2‐2

Introduction .............................................................................................................. U2‐2

Section Learning Outcomes .................................................................................... U2‐2

Study Plan ................................................................................................................. U2‐2

Sample Question from Section 2.1 ......................................................................... U2‐3

Further Practice Section 2.1 ..................................................................................... U2‐4

2.2: Graphs of Equations ............................................................................................ U2‐5

Introduction .............................................................................................................. U2‐5

Section Learning Outcomes .................................................................................... U2‐5

Study Plan ................................................................................................................. U2‐5

Sample Questions for Section 2.2 ........................................................................... U2‐6

Further Practice Section 2.2 ..................................................................................... U2‐8

2.3: Lines ........................................................................................................................ U2‐9

Introduction .............................................................................................................. U2‐9

Section Learning Outcomes .................................................................................... U2‐9

Study Plan ................................................................................................................. U2‐9

Sample Questions for Section 2.3 ......................................................................... U2‐10

Further Practice Section 2.3 ................................................................................... U2‐14

2.4: Definition of Function ...................................................................................... U2‐15

Introduction ............................................................................................................ U2‐15

Section Learning Outcomes .................................................................................. U2‐15

Study Plan ............................................................................................................... U2‐15

Section 2.4: Definition of Function ....................................................................... U2‐16

Further Practice Section 2.4 ................................................................................... U2‐17

2.5: Graphs of Functions .......................................................................................... U2‐19

Introduction ............................................................................................................ U2‐19

Section Learning Outcomes .................................................................................. U2‐19

Study Plan ............................................................................................................... U2‐19

Sample Questions from Section 2.5 ..................................................................... U2‐20

Further Practice for Section 2.5 ............................................................................. U2‐22

2.6: Quadratic Functions .......................................................................................... U2‐23

Introduction ............................................................................................................ U2‐23

Section Learning Outcomes .................................................................................. U2‐23

Study Plan ............................................................................................................... U2‐23

Sample Questions for Section 2.6 ......................................................................... U2‐24

Further Practice Section 2.6 ................................................................................... U2‐26

2.7: Operations on Functions................................................................................... U2‐27

Introduction ............................................................................................................ U2‐27

Section Learning Outcomes .................................................................................. U2‐27

Study Plan ............................................................................................................... U2‐27

Sample Questions from Section 2.7 ..................................................................... U2‐28

Further Practice Section 2.7 ................................................................................... U2‐30

MATH 1001: Pre-Calculus Mathematics U2-1

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Unit 2: Functions and Graphs This unit corresponds to Chapter 2 of the textbook. In Unit 2 you will be covering the following sections in your textbook:

 2.1: Rectangular Coordinate Systems

 2.2: Graphs of Equations

 2.3: Lines

 2.4: Definition of Function

 2.5: Graphs of Functions

 2.6: Quadratic Functions

 2.7: Operations on Functions

Introduction In many situations of practical interest we see cases in which two variables are related in such a way that the value of one is completely determined by that of the other. For example, since a particle cannot be in two places at the same time the history of a particle’s motion determines a unique position for each time.

Relationships of this sort are called functions. They are particularly important as they are the basis for virtually all of the predictive power of the sciences. Even in the absence of specific formulas the language of functions can be used to describe what depends on what and this is often at least as important as the details of exactly how the dependence works. And when one does want to talk about how strongly one quantity depends on another (as studied in calculus) then it must in fact be dependent and so be determined by a function. In fact, the language of functions is used to express most of the ideas of calculus and in all of its areas of application.

Learning Outcomes This unit is where you will develop and confirm your understanding of the mathematical notion of a function and of the notation and language used to describe functions. Specific learning outcomes are listed at the beginning of each section in the unit.

U2-2 Unit 2: Functions and Graphs

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2.1: Rectangular Coordinate Systems

Section Learning Outcomes After completing this section you should be able to:

 Use a Cartesian coordinate system to plot and identify points in a plane.

 Express distance and slope between two points in terms of their Cartesian coordinates.

Study Plan 1. Read section 2.1 of the textbook. You should already know most of what is in this

section of the textbook. But graphing is so important for the rest of this course, and for calculus, that you should read it very carefully and make sure to remind yourself of any parts you have forgotten. Keep a pencil and paper at hand, and be sure to check your understanding by working through the examples as you go.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

Study Notes and Sample Questions Not all of the material presented in each section of the textbook is required for this course and some topics are more important than others. To master the required material in this section, review the following topics and examples and the supplemental “Sample Questions from Section 2.1”.

MATH 1001: Pre-Calculus Mathematics U2-3

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Section 2.1: Rectangular Coordinate Systems

Topic Page, Example Number

Distances Between Points Pages 82, 83

Page 83—Example 1

Page 83—Example 2

Midpoint Formula Page 85

Page 85—Example 5

Sample Question from Section 2.1

Page 88—q. 18 Show that ( 4, 1) , (0, 2) , (6,1)A B C     and (2, 2)D  are vertices of a parallelogram.

Solution:

Show that opposite sides have equal lengths.

     

   

2 2

4 2 1 2

36 9

A D A dd A D x x y y   

     

 



             

2 2 2 2, 0 6 2 1 45

, , B C B Cd B C x x y y

d A D d B C

         

 

     2 2, 4 0 1 2 17d A B        and

     2 2, 6 2 1 2 17d C D          , ,d A B d C D

Since opposite sides have equal lengths, ABCD is a parallelogram.

U2-4 Unit 2: Functions and Graphs

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Further Practice Section 2.1 Check your understanding, and improve your speed, by working through some of the exercises on pages 88–89 of the textbook.

Do enough of the odd‐numbered questions of each type to convince yourself that you can get the right answers. Note that, the answers are at the back of the textbook and complete worked‐out solutions are in the Student Solutions Manual––but try to avoid looking at answers or solutions until you have made your own best effort.

As a minimum, start with questions 1, 3, 5, 7, 9, 15 and 17 from Section 2.1. When done, compare your solutions to those in the Student Solutions Manual.

MATH 1001: Pre-Calculus Mathematics U2-5

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2.2: Graphs of Equations

Introduction The use of Cartesian coordinates allows us to bring together algebra and geometry in a way that works to our advantage in both. By expressing geometric ideas in terms of coordinates, we can use algebraic techniques to prove geometric theorems and calculate properties of geometric objects, and by graphing equations and inequalities we can use our physical and geometric intuition to understand their algebraic properties.

Section Learning Outcomes After completing this section you should be able to:

 Find the equation of a straight line specified in various ways.

 Plot graphs of linear and quadratic equations and identify the equations from given or specified graphs.

 Find equations of circles with specified centre and radius.

 Find intersection points of pairs of lines and circles.

 Apply these skills in a wide variety of practical situations.

Study Plan 1. Read section 2.2 of the textbook. Most of what is in this section of the textbook

should already be known to you. But graphing is so important for the rest of this course and for calculus that you should read it very carefully and make sure to remind yourself of any parts you have forgotten. Keep a pencil and paper at hand, and check your understanding by working through the examples as you go.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

U2-6 Unit 2: Functions and Graphs

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Section 2.2: Graphs of Equations

Topic Page, Example Number

Graphing by Plotting Points Pages 89–92

Page 90—Example 1

Page 91—Example 2

Finding x and y Intercepts Page 92—Example 3

Symmetry of Graphs Pages 94, 95

Page 95—Example 5

Page 95—Example 6

Standard Equation of a Circle Page 96

Page 96—Example 7

Completing the Square Technique Page 97

Page 97—Example 8

Page 97—Example 9

Sample Questions for Section 2.2 Sketch the graph of the equation and label the x and y intercepts.

3 2y x 

Solution:

x ‐intercept: 20 3 2 0 3

y x x      

y ‐intercept: 0 3(0) 2 2x y y     

So the intercepts are 2: , 0 3

x     

and  : 0, 2y

MATH 1001: Pre-Calculus Mathematics U2-7

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NOTE: From the slope‐intercept form, y mx b  , the slope is 3m  , which is positive. So the line rises as we move from the left to the right on the x axis.

Page 102—q. 10 Sketch the graph and label the xand y  intercepts.

22x y 

Solution:

x ‐intercept: 0 2(0) 0 0y x x      

y ‐intercept: 20 0 2 0x y y     

So the graph passes through the origin (0, 0) . The graph is symmetric with respect to xaxis .

U2-8 Unit 2: Functions and Graphs

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Page 103—q. 58 Find the centre and the radius of the circle with the given equation.

2 2 4 6 16 0x y x y    

Solution:

Group the terms in x and the terms in y and subtract 16 from both sides:

   2 24 6 16x x y y     Complete the perfect square in each parenthesis on the LHS and add the same amount s to the RHS:

   2 24 4 6 9 16 4 9x x y y            2 22 3 3x y    

This is not a circle since 2r cannot equal 3 .

Further Practice Section 2.2 Check your understanding, and improve your speed, by working through some of the exercises on pages 102–103 of the textbook.

As a minimum, you should do questions 1, 3, 5, 7, 15, 17, 21, 35, 37, 45, 47, 51, 57 and 67 from Section 2.2. When done, compare your solutions to those in the Student Solutions Manual.

MATH 1001: Pre-Calculus Mathematics U2-9

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2.3: Lines

Introduction The topic of linear equations gets a whole section of the textbook to itself. This is partly because of the significance of straight lines in geometry and partly because of the convenience of linear models for approximating more general relationships. In fact, one of the main ideas of calculus is to use linear approximations to examine small parts of more complicated graphs. A recent survey of post‐secondary math instructors in British Columbia identified an understanding of the concept of slope of a straight line as the single most important prerequisite skill for subjects as diverse as calculus and statistics.

Section Learning Outcomes After completing this section you should be able to:

 Find the equation of a straight line specified in various ways.

 Apply these skills in a wide variety of practical situations.

Study Plan 1. Read section 2.3 of the textbook. You should already know most of what is in this

section of the textbook. But graphing linear equations is so important for the rest of this course and for calculus that you should read it very carefully and make sure to remind yourself of any parts you have forgotten. Keep a pencil and paper at hand and check your understanding by working through the examples as you go.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

U2-10 Unit 2: Functions and Graphs

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Section 2.3: Lines

Topic Page, Example Number

Slope Page 105

Page 106—Example 1

Horizontal and Vertical Lines Page 108

Page 108—Example 3

Point‐Slope Form Page 108

Page 109—Example 4

Slope‐Intercept Form Page 109

Page 109—Example 5

General Form Page 110

Page 110—Example 6

Parallel and Perpendicular Lines Pages 111–113

Page 111—Example 7

Page 112—Example 8

Pages 113, 114—Example 10

Sample Questions for Section 2.3 Use slopes to show that the points are vertices of the specified trapezoid.

(2, 3), (5, 1), (0, 6), ( 6, 2)A B C D      

MATH 1001: Pre-Calculus Mathematics U2-11

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Solution:

Show that the slopes of one pair of opposite sides are equal.

1 3 4 5 2 3AB

m     

2 ( 6) 8 4 6 0 6 3CD

m        

Page 117—q. 22, 28 Find a general form of an equation of the line through the point A that satisfies the given condition.

q. 22

( 4, 2)A   and the line is

(a) parallel to the xaxis

(b) perpendicular to the xaxis

Solution:

(a) A line parallel to the x ‐axis is horizontal, so it has the form y c . Since it passes through ( 4, 2) , it must be 2y  .

(b) A line perpendicular to the y‐axis is vertical, so it has the form x c . Since it passes through ( 4, 2) , it must be 4x   .

Page 117, q. 28 ( 1, 6)A   x intercept = 5

Solution:

The coordinates of the x ‐intercept are (5, 0) and so the slope of the line containing this point and A is

1 2

6 0 1 1 5

y ym x x  

      

U2-12 Unit 2: Functions and Graphs

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We write the equation of the line through ( 1, 6)A   and the x intercept =  5,0 using the point‐slope form of a line:

 1 1y y m x x  

 6 1 ( 1) 6 1( 1)y x y x          which can also be written as

5y x   (slope‐intercept form) OR 5x y  (general form)

Page 118—q. 50

Find an equation of the line that is tangent to the circle 2 2 25x y  at the point  3, 4P  .

Solution:

The line through the origin (0, 0)O  and (3, 4)P  is perpendicular to the desired line.

4 0 4 3 0 3

P O OP

P O

y ym x x

    

 

 slope of desired line: 1 3 4OP

m m

   

MATH 1001: Pre-Calculus Mathematics U2-13

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The equation of the desired line, in point‐slope form is

 34 3 4

y x    or 3 25 4 4

y x  

Page 118—q. 58 Loan Repayment

A college student receives an interest‐free loan of $8,250 from a relative.

The student will repay $125 per month until the loan is paid off.

a) Express the amount P (in dollars) remaining to be paid in terms of time t (in months).

b) After how many months will the student owe $5,000?

c) Sketch on the tP  plane, a graph that shows the relationship between P and t for the duration of the loan.

Solution:

a) Using the slope‐intercept form: P mt b 

At the beginning ( 0)t  , the amount remaining is $8,250 so 8,250b 

 8,250P mt  for any t

After one payment, i.e. for 1t  , the amount remaining decreases by $125 and so $8,125P  .

8,125 8,250 125 125 8,250m m P t        

b) $5,000 ?P t  

5,000 125 8,250 125 3,250 26 monthst t t        

c) To sketch the graph of the line 125 8,250,P t   we need the intercepts with the axes:

t—intercept:

8,2500 125 8,250 0 66 125

P t t      

P—intercept:

0 8,250t P  

So the intercepts are (66, 0) and (0, 8,250) .

U2-14 Unit 2: Functions and Graphs

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Further Practice Section 2.3 Check your understanding, and improve your speed, by working through some of the exercises on pages 116–120 of the textbook.

As a minimum, you should do questions 1, 3, 9, 11, 15, 17, 19, 21, 23, 29, 31, 33, 41, 43, 45, 49, 57, and 59 from Section 2.3. When done, compare your solutions to those in the Student Solutions Manual.

MATH 1001: Pre-Calculus Mathematics U2-15

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2.4: Definition of Function

Introduction As mentioned in the “Introduction” to this unit, a function is a particular kind of relationship. In fact, the basic idea can be illustrated in terms of human relationships.

Every person has a unique birth mother. So, if we have a particular person, p, in mind then there is only one person that we can mean by “the birth mother of p.” This person, who we’ll call ( )M p , is completely determined once we know p, but the reverse is not true. If we identify a mother, she may have several children, so the phrase “child of m” does not always determine a unique person.

Note that in this mathematical sense the mother “depends on” and “is a function of” the child.

Section Learning Outcomes After completing this section you should be able to:

 Define the term “function” and identify whether or not a given relationship is a function.

 Define the domain, range, and graph of a function and identify them for specific examples.

 Interpret expressions written in terms of function notation.

Study Plan 1. Read section 2.4 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

U2-16 Unit 2: Functions and Graphs

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Section 2.4: Definition of Function

Topic Page, Example Number

Definition of Function Page 120, 121

Page 122—Example 1

Page 122—Example 2

Vertical Line Test Page 124

Increasing, Decreasing and Constant Functions

Page 125

Domain and Range Page 126—Example 4

The Difference Quotient Page 127

Page 127—Example 5

Linear Functions Pages 127–129

Page 128—Example 6

Page 128—Example 7

Sample Questions for Section 2.4 If a and h are real numbers, find:

(a)  f a (b)  f a

(e)    f a f h (f)     0if f a h f a h

h  



where   3 4f x x  for any x .

Solution:

(a) If x a then   3 4f a a 

(b) If x a  then    3 4 3 4f a a a     

MATH 1001: Pre-Calculus Mathematics U2-17

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(d) If x a h  then    3 4 3 4 4f a h a h a h      

(e)        3 4 3 4 6 4 4f a f h a h a h       

(f)        3 4 4 3 4 4 4f a h f a a h a h h h h

          

Page 132—q. 28

Find the domain of f where   2 4 3

4 xf x

x 

 

Solution:

34 3 0 4

x x    and 2 4 0 2x x    

So the domain is all real numbers that satisfy both conditions.

 3 , 2 2, 4     

Section 2.4 example

Simplify the difference quotient    2 2 0iff h f h h

  

for the function   22 3f x x  

Solution:

     

2 2 2

2 2 2 3 2 4 4 3 8 8 2 3

2 8 5

2 2 2 3 2 4 3 8 3 5

f h h h h h h

h h

              

   

          

     22 2 2 8 5 5 2 8f h f h h h h h

          

Further Practice Section 2.4 Check your understanding, and improve your speed, by working through some of the exercises on pages 131–135 of the textbook. Do enough of the odd numbered questions of each type to convince yourself that you can get the right answers. Note that the answers are at the back of the textbook and complete worked‐out solutions

U2-18 Unit 2: Functions and Graphs

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are in the Student Solutions Manual––but try to avoid looking at answers or solutions until you have made your own best effort.

As a minimum, you should do questions 1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 27, 33, 35, 37, 39, 47, 49, 51, 53, and 67 from Section 2.4. When done, compare your solutions with those in the Student Solutions Manual.

MATH 1001: Pre-Calculus Mathematics U2-19

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2.5: Graphs of Functions

Introduction We have already seen that having the graph of a function allows us to quickly estimate its values and behaviours such as whether its output will increase or decrease when the input is changed. In this section we look at some techniques that will help you to sketch and understand the graphs of functions. These include the use of symmetry, and recognizing how the graph of a given function might be related to a simpler or better known one.

Section Learning Outcomes After completing this section you should be able to:

 Identify and use the symmetry properties of even and odd functions.

 Determine the graphs of equations written in terms of functions and write equations for graphs obtained by transformations of a given graph.

Study Plan 1. Read section 2.5 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

U2-20 Unit 2: Functions and Graphs

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Section 2.5: Graphs of Functions

Topic Page, Example Number

Even and Odd Functions Page 136

The Absolute Value Function Page 137—Example 2

Transformations Pages 138–142

Vertical Shifts Pages 138, 139

Page 138—Example 3

Horizontal Shifts Pages 139, 140

Page 139—Example 4

Vertical Stretches and Compressions Page 140

Page 140—Example 5

Horizontal Stretches and Compressions

Pages 141, 142

Page 141—Example 7

Reflections Pages 140, 141

Page 140—Example 6

Piecewise‐Defined Functions Page 142

Page 142—Example 8

Page 145—Example 11

Sample Questions from Section 2.5

Page 147—q. 18 Sketch, on the same coordinate plane, the graphs of f for the given values of c (make use of symmetry, shifting, sketching, compressing or reflecting).

  29 3,0,2f x x c c    

MATH 1001: Pre-Calculus Mathematics U2-21

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Solution:

When 0c  , 2( ) 9f x x  which is the equation of the upper half of a circle of

radius 3 centred at the origin. For 3, 2c   shift 29y x  down 3, up 2, respectively.

Page 148—q. 34

Explain how the graph of the function 3 ( 1)y f x   3 1y f x  compares to the graph of  y f x . For example, for the equation  2 3y f x  , the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2.

Solution:

The graph of f is shifted 1 unit to the right and stretched vertically by a factor of 3.

Page 149—q. 51 Sketch the graph of f

  3 2 1

1 ( 1 1) 3 1

if if i.e.

x x f x x x x

x x

           

Solution:

This is a piecewise‐defined function. So the graph is a combination of the graphs of the line 2y x  for values of 1x   , the power function 3y x for values of x between –1 and 1, and the line 3y x   for values of 1x  .

U2-22 Unit 2: Functions and Graphs

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Further Practice for Section 2.5 Check your understanding and improve your speed by working through some of the exercises on pages 147–150 of the textbook. Do enough of the odd‐numbered questions of each type to convince yourself that you can get the right answers. Note that the answers are at the back of the textbook book and complete worked‐out solutions are in the Student Solutions Manual—but try to avoid looking at the answers or solutions until you have made your own best effort.

As a minimum, you should do questions 1, 3, 9, 13, 15, 17, 27, 29, 33, 35, 39, 41 (a), (e), 43, 47, 49, 51, 55, 57, 59, 65 (a), (c), and 67 from Section 2.5. When done, compare your answers to those in the student solutions guide.

MATH 1001: Pre-Calculus Mathematics U2-23

TRU Open Learning

2.6: Quadratic Functions

Introduction A shoe retailer finds that she can sell (on average) 50 pairs per week of a certain style when the price is $50 per pair, but that for each $5 she raises the price she loses 4 sales.

The store owner’s weekly revenue is obtained by multiplying the price she charges by the number of pairs she sells and one goes down as the other goes up so the combined effect is not particularly obvious.

One of the things you will learn in this section is how to use a quadratic function to model the above situation so that you can tell the retailer what price to charge in order to maximize her weekly revenue.

Section Learning Outcomes After completing this section you should be able to:

 Solve quadratic equations.

 Graph quadratic functions and determine the vertex and the x and y intercepts from the general form.

 Use the “Completing the Square” technique (page 46 and page 97) to convert a quadratic equation to standard form.

 Use the standard form of a quadratic function to locate the vertex and to find the maximum or minimum value of the function.

 Recognize applied situations involving these functions and use the above skills to solve problems arising from such situations.

Study Plan 1. Read section 2.6 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

U2-24 Unit 2: Functions and Graphs

TRU Open Learning

Section 2.6: Quadratic Functions

Topic Page, Example Number

Definition and Graph Pages 151, 152

Page 152—Example 1

Standard Equation Page 153

Page 152—Example 2

Page 153—Example 3

Page 154—Example 4

Vertex Formula Page 155

Page 155—Example 6

Maximum and Minimum Values Page 156

Page 158—Examples 8, 9

Sample Questions for Section 2.6 For   2 6f x x x   do the following:

(a) Use the quadratic formula to find the zeros of f .

(b) Find the maximum or minimum value of  f x .

Solution:

Generally:   2f x ax bx c  

a)   2 6 1 6 0f x x x a b c        

To find the zeros of f solve   20 6 0f x x x     

6 36 0 6;0 2

x     

b) The coefficient of 2x in this case is 1a   . Since 0a  , the graph is a parabola that opens downward. Therefore the highest point on the parabola

is at the vertex , 2 2 b bV f a a

        

MATH 1001: Pre-Calculus Mathematics U2-25

TRU Open Learning

6 3 2 2( 1) b a

     

 and 2( 3) ( 3) 6( 3) 9f        

So ( 3, 9)V   and the maximum value of f is 9MAXy 

c) The x—intercepts are ( 6, 0) and (0, 0) and the vertex is ( 3, 9) .

Page 162—q. 42

Flight of a Projectile

An object is projected vertically upward with an initial velocity of 0v ft/sec, and its distance  s t in feet above the ground after t seconds is given by the formula   216 os t t v t  

a) If the object hits the ground after 12 seconds, find its initial velocity 0v .

b) Find its maximum distance above the ground.

Solution:

a) When the object hits the ground, ( ) 0s t  2

0 016(12) (12) 0 192v v    

So the initial velocity is 192 ft / sec .

b)   216 192s t t t   16 192a b  

Since 0a  the graph opens downward and ( )s t has a maximum value

when 192 6 2( 16)

t    

and 2(6) 16(6) 192(6) 576s     .

So, the maximum distance is 576 ft .

U2-26 Unit 2: Functions and Graphs

TRU Open Learning

Further Practice Section 2.6 Check your understanding, and improve your speed, by working through some of the exercises on pages 160–163 of the textbook.

Do enough of the odd‐numbered questions of each type to convince yourself that you can get the right answers. Note that the answers are at the back of the textbook and complete worked‐out solutions are in the Student Solutions Manual––but try to avoid looking at answers or solutions until you have made your own best effort.

As a minimum, you should try questions 1, 5, 9, 13, 19, 23, 43, and 45 from Section 2.6. When done, compare your solutions with those in the Student Solutions Manual.

MATH 1001: Pre-Calculus Mathematics U2-27

TRU Open Learning

2.7: Operations on Functions

Introduction If two branches of a business each generate revenue at an earnings rate that varies over time then the total earnings rate at any given time will be obtained by adding the values of these two functions. If the number of customers and their average monthly expenditure at a certain business are given functions of time the monthly revenue of the business will be a function of time, the value of which is obtained by multiplying the values of those two given functions. If the air temperature depends on altitude according to a given function and a balloon is rising so that its altitude is given by another function, then the function giving air temperature outside the balloon as a function of time is obtained by combining these two—but in a different way from either the addition of revenues or multiplication of price times number in the first two examples.

Section Learning Outcomes After completing this section you should be able to:

 Understand and use the language and notation used to describe algebraic combinations of functions.

 Evaluate, graph, and identify domains of algebraic combinations of functions identified in various ways (by formulas, tables, graphs, etc.).

 Understand and use the language and notation used to describe composition of functions.

 Evaluate, graph and identify domains of composite functions using information about the components that is given in various ways (formulas, tables, graphs, etc.).

 Recognize a function description as equivalent to a composition of simpler functions.

Study Plan 1. Read section 2.7 of the textbook.

2. Read the following “Study Notes and Sample Questions” section to prepare for the practice exercises.

3. Follow the instructions regarding “Further Practice.”

U2-28 Unit 2: Functions and Graphs

TRU Open Learning

Study Notes and Sample Questions Not all of the material in each section of the textbook is required for this course and some topics are more important than others. To master the required material in this section, review the following topics and examples and the supplemental “Sample Questions from Section 2.7”.

Section 2.7: Operations on Functions

Topic Page, Example Number

Operations with Functions Page 165

Page 165—Example 1

Page 166—Example 2

Composite Function Page 166

Page 167—Example 3

Page 168—Example 4

Page 168—Example 5

Composite Function Form Page 169—Example 7

Sample Questions from Section 2.7

Page 172—q. 16

For     25 7 3 2f x x g x x x     find the following:

(a)   ( ) ( ( ))f g x f g x  (b)   ( ) ( ( ))g f x g f x 

Solution:

(a)      2 2 2( ) 3 2 5 3 2 7 15 5 3f g x f x x x x x x         

(b)        2 2( ) 5 7 3 5 7 5 7 2 75 215 156g f x g x x x x x         

For      22 ( 2) 15 2 5 2 3 73x f g         

MATH 1001: Pre-Calculus Mathematics U2-29

TRU Open Learning

(d) Using (b)   2( ) 75 215 156g f x x x   

For      23 (3) 75 3 215 3 156 186x g f     

Page 172—q. 32

For     3 2

xf x g x x x

  

find the following:

(a)   ( )f g x and the domain of f g .

(b)   ( )g f x and the domain of g f .

Solution:

(a)   3 3

3 3 3( ) ( ( )) 3 3 2 3 2 3 22

xx xf g x f g x f xx x x x x x

              

The domain of g is all 0x  and the domain of f is all 2x  .

The domain of f g is all x such that 0x  and ( ) 2g x  . 3 3( ) 2 2

2 g x x

x     

So the domain of f g is all 30, 2

x  .

(b)

 

   

3 3 2( ) ( ( )) 2 1

2 3 2 3 2 3 6

x xg f x g f x g xx x x

x x x x x x

        

     



The domain of g f is all x such that 2x  and ( ) 0f x  .

( ) 0 0f x x   . So the domain of g f is all 0, 2x  .

Page 172—q. 35

Solve the equation   ( ) 0f g x  for the functions

   2 2 3f x x g x x   

U2-30 Unit 2: Functions and Graphs

TRU Open Learning

Solution:

  ( ) 0 ( ( )) 0 ( 3) 0f g x f g x f x      2 2( 3) 2 0 ( 3) 2 3 2

3 2

x x x

         

   

Page 174—q. 56 Find a composite function form for y .

24 1y x  

Solution:

Let 2 1 4u x y u    

Further Practice Section 2.7 Check your understanding and improve your speed by working through some of the exercises on pages 171–174 of the textbook.

Do enough of the odd‐numbered questions of each type to convince yourself that you can get the right answers. Note that, as usual, the answers are at the back of the textbook and complete worked‐out solutions are in the Student Solutions Manual–– but try to avoid looking at answers or solutions until you have made your own best effort.

As a minimum, you should do questions 1, 3, 7, 9, 11, 13, 21, 23, 29, 31, 33, 35, 37, 53, 55, and 57 from Section 2.7. When done, compare your solutions with those in the Student Solutions Manual.