Radical and power functions root and radical functions
Precalculus
Version 4 −ϵ
by
September 22, 2017
Table of Contents
1.1.2 Algebraic Representations of Functions . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Geometric Representations of Functions . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.2.3 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.3.1 Graphs of Absolute Value Functions . . . . . . . . . . . . . . . . . . . . . . . . 74
1.3.2 Graphical Solution Techniques for Equations and Inequalities . . . . . . . . . . . 84
1.4.2 Inequalities involving Quadratic Functions . . . . . . . . . . . . . . . . . . . . . 111
1.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.1.2 Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
2.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
2.3 Real Zeros of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
2.3.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
2.4 Complex Zeros and the Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . 197
3.1.1 Laurent Monomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
3.1.2 Local Behavior near Excluded Values . . . . . . . . . . . . . . . . . . . . . . . . 218
3.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
3.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
4.1 Root and Radical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.1.1 Root Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
4.2.1 Rational Number Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
4.2.2 Real Number Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
4.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Table of Contents v
5.2 Function Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
5.2.1 Difference Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
5.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
5.4 Transformations of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
5.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
5.4.6 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
5.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
5.6.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
6.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
6.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
6.4 Equations and Inequalities involving Exponential Functions . . . . . . . . . . . . . . . . 527
6.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
6.6 Applications of Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . 551
vi Table of Contents
7 The Conic Sections 579
7.1 Introduction to Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
7.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
7.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
7.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
7.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
8.2 Systems of Linear Equations: Augmented Matrices . . . . . . . . . . . . . . . . . . . . . 661
8.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
8.4 Systems of Linear Equations: Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . 692
8.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701
8.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
8.5.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
8.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738
Table of Contents vii
9 Sequences and the Binomial Theorem 755
9.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755
9.2.2 Extensions to Calculus: Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777
9.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782
9.4 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794
9.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805
10.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816
10.1.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818
10.3 Graphs of Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845
10.3.1 Applications of Sinusoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855
10.4.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 880
10.5 Graphs of Secant, Cosecant, Tangent, and Cotangent Functions . . . . . . . . . . . . . 883
10.5.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 900
viii Table of Contents
11.2 More Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916
11.2.1 Sinusoids, Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929
11.3.2 Inverses of Secant and Cosecant: Calculus Friendly Approach . . . . . . . . . . 956
11.3.3 Calculators and the Inverse Circular Functions. . . . . . . . . . . . . . . . . . . . 960
11.4.1 Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008
11.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015
12.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034
12.1.3 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037
12.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064
12.3.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068
13 Polar Coordinates and Parametric Equations 1087
13.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087
13.2.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133
Table of Contents ix
13.4.1 Rotation of Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165
13.4.2 The Polar Form of Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173
13.5.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202
A Algebra Review 1209
A.1.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223
A.2 Real Number Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226
A.3.2 Distance in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1251
A.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257
A.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271
A.4.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273
A.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297
A.6.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298
A.7.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307
x Table of Contents
A.8.4 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317
A.9 Basic Factoring Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318
A.10.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1342
A.10.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343
A.12.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365
A.12.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367
A.14 Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383
A.14.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385
B.1.2 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399
B.2 Right Triangle Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401
C.2 Why Pi is Actually a Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416
C.3 Cauchy’s Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417
Carl and I are natives of Northeast Ohio. We met in graduate school at Kent State University in 1997. I finished my Ph.D in Pure Mathematics in August 1998 and started teaching at Lorain County Community College in Elyria, Ohio just two days after graduation. Carl earned his Ph.D in Pure Mathematics in August 2000 and started teaching at Lakeland Community College in Kirtland, Ohio that same month. Our schools are fairly similar in size and mission and each serves a similar population of students. The students range in age from about 16 (Ohio has a Post-Secondary Enrollment Option program which allows high school students to take college courses for free while still in high school.) to over 65. Many of the “non-traditional”students are returning to school in order to change careers. A majority of the students at both schools receive some sort of financial aid, be it scholarships from the schools’ foundations, state-funded grants or federal financial aid like student loans, and many of them have lives busied by family and job demands. Some will be taking their Associate degrees and entering (or re-entering) the workforce while others will be continuing on to a four-year college or university. Despite their many differences, our students share one common attribute: they do not want to spend $200 on a College Algebra book.
The challenge of reducing the cost of textbooks is one that many states, including Ohio, are taking quite seriously. Indeed, state-level leaders have started to work with faculty from several of the colleges and universities in Ohio and with the major publishers as well. That process will take considerable time so Carl and I came up with a plan of our own. We decided that the best way to help our students right now was to write our own College Algebra book and give it away electronically for free. We were granted sabbaticals from our respective institutions for the Spring semester of 2009 and actually began writing the textbook on December 16, 2008. Using an open-source text editor called TexNicCenter and an open-source distribution of LaTeX called MikTex 2.7, Carl and I wrote and edited all of the text, exercises and answers and created
| written by Dr. Previts were added and the typographical errors found by our students and others were corrected. On December 10, 2009, Version√2 was released. The book remains free for download at our |
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Critics of the Open Educational Resource movement might quip that “open-source is where bad content goes to die,” to which I say this: take a serious look at what we offer our students. Look through a few sections to see if what we’ve written is bad content in your opinion. I see this open-source book not as something which is “free and worth every penny”, but rather, as a high quality alternative to the business as usual of the textbook industry and I hope that you agree. If you have any comments, questions or concerns please feel free to contact me at jeff@stitz-zeager.com or Carl at carl@stitz-zeager.com.
1.1 Functions and their Representations
1.1.1 Functions as Mappings
The grammar here ‘from A to B ’ is important. Thinking of a function as a process, we can view the elements of the set A as our starting materials, or inputs to the process. The function processes these inputs according to some specified rule and the result is a set of outputs - elements of the set B. In terms of inputs and outputs, Definition 1.1 says that a function is a process in which each input is matched to one and only one output.
2 Introduction to Functions
| N (inputs) | f | T (outputs) | g | N (outputs) | |||
|---|---|---|---|---|---|---|---|
| White Paw | cat lizard turtle |
White Paw
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|||||
| Cooper | |||||||
| Bingo | |||||||
| Kennie | |||||||
|
|---|
1.1 Functions and their Representations 3
Some remarks about Definition 1.2 are in order. First, and most importantly, the notation ‘f(a)’ in Definition 1.2 introduces yet another mathematical use for parentheses. Parentheses are used in some cases as grouping symbols, to represent ordered pairs, and to delineate intervals of real numbers. More often than not, the use of parentheses in expressions like ‘f(a)’ is confused with multiplication. As always, paying attention to the context is key. If f is a function and ‘a ’ is in the domain of f, then ‘f(a)’ is the output from f when you input a. The diagram below provides a nice generic picture to keep in mind when thinking of a function as a mapping process with input ‘a ’ and output ‘f(a)’.
In the preceding pet example, the symbol f(Bingo), read ‘f of Bingo’, is asking what type of pet Bingo is, so f(Bingo) = lizard. The fact that f is a function means f(Bingo) is unambiguous because f matches the name ‘Bingo’ to only one pet type, namely ‘lizard’. In contrast, if we tried to use the notation ‘g(cat)’ to indicate what pet name g matched to ‘cat’, we have two possibilities, White Paw and Cooper, with no way to determine which one (or both) is indicated.
Continuing to apply Definition 1.2 to our pet example, we find that the domain of the function f is N, the set of pet names. Finding the range takes a little more work, mostly because it’s easy to be caught off guard by the notation used in the definition of ‘range’. The description of the range as ‘{f(a) | a ∈A}’ is an example of ‘set-builder’ notation. In English, ‘{f(a) | a ∈A}’ reads as ‘the set of f(a) such that a is in A’. In other words, the range consists of all of the outputs from f - all of the f(a) values - as a varies through each of the elements in the domain A. Note that while every element of the set A is, by definition, an element of the domain of f, not every element of the set B is necessarily part of the range of f.2
not have been the entire codomain U. 3If instead of mapping N into T, we could have mapped N into U = {cat, lizard, turtle, dog} in which case the range of f would
4 Introduction to Functions
While the concept of a function is very general in scope, we will be focusing primarily on functions of
real numbers because most disciplines use real numbers to quantify data. Our next example explores a
| time (hours after 6 a.m.) | outdoor temperature in degrees Fahrenheit | |
|---|---|---|
| 0 | 64 | |
| 2 | 67 | |
| 4 | 75 | |
| 6 | 80 | |
| 8 | 83 | |
| 10 | 83 | |
| 12 | 82 |
2. Is time a function of the outdoor temperature? Explain.
3. Let f be the function which matches time to the corresponding recorded outdoor temperature.
| • f(2) | • f(4) | • f(2 + 4) |
|---|
(c) State the range of f. What is lowest recorded temperature of the day? The highest?
4These adjectives stem from the fact that the value of t depends entirely on our (independent) choice of n.
1. The outdoor temperature is a function of time because each time value is associated with only one recorded temperature.
2. Time is not a function of the outdoor temperature because there are instances when different times are associated with a given temperature. For example, the temperature 83 corresponds to both of the times 8 and 10.
| 3. | (a) |
|---|
• Using results from above we see that f(2) + f(4) = 67 + 75 = 142. When adding f(2) + f(4), we are adding the recorded outdoor temperatures at 8 a.m. (2 hours after 6 a.m.) and 10 a.m. (4 hours after 6 AM), respectively, to get 142◦F.
• We compute f(2) + 4 = 67 + 4 = 71. Here, we are adding 4◦F to the outdoor temperature recorded at 8 a.m..
A few remarks about Example 1.1.1 are in order. First, note that f(2 + 4), f(2) + f(4) and f(2) + 4 all work out to be numerically different, and more importantly, all represent different things.7One of the common mistakes students make is to misinterpret expressions like these, so it’s important to pay close attention to the syntax here.
Next, when solving f(t) = 83, the variable ‘t ’ is being used as a convenient ‘dummy’ variable or placeholder in the sense that solving f(t) = 83 produces the same solutions as solving f(x) = 83, f(w) = 83, or even f(?) = 83. All of these equations are asking for the same thing: what inputs to f produce an output of 83. The choice of the letter ‘t ’ here makes sense since the inputs are time values. Throughout the text, we will endeavor to use meaningful labels when working in applied situations, but the fact remains that the choice of letters (or symbols) is completely arbitrary.
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|---|
Definition 1.3 is an example where the name of the function, f, is being used almost synonymously with its outputs in that when we speak of ‘the minimum and maximum of the function f ’ we are really talking about the minimum and maximum values of the outputs f(x) as x varies through the domain of f. Thus we say that the maximum of f is 83 and the minimum of f is 64 when referring to the highest and lowest recorded temperatures in the previous example.
1.1.2 Algebraic Representations of Functions
• Step 2: multiply the result of Step 1 by 3
• Step 3: subtract 1 from the result of Step 2.
