Precalculus Notes

Remkes Kooistra

September 10, 2018

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 InternationalLicense.

This document seeks to provide a concise overview of the technical skills required for university levelcalculus. Many students have asked me: What do I need to know going into your first year calculuscourse? Here is the answer.

I’ve written this document to include everything I expect (or hope) has been mentioned in a typi-cal high-school cirriculum. In the calculus class, I will assume that the majority of this material isknown. That said, there are a number of section which I will take time to review during class. Theseinclude various new notations, Cartesian geometry, equations of linear, types of functions, propertiesof function, composition of functions and inversion of functions.

Since mathematics is so interconnected, the order of topics is not entirely linear. For example, by sheernecessity, we refer to functions before the functions section. The best use of this document is holistic,so that the various pieces can all inform each other.

Some of the notations may be different from what students have learned in high school. These notationsare consistent with the notations I use in calculus courses, as well as with general usage in universitylevel mathematics. I will carefully define the new notations as I introduce them.

In addition to these notes, I have made several other documents available to students. These include acomprehensive reference sheet, a library of functions with graphs, and notes on vector geometry. Theseadditional reference documents, while designed to work with the calculus course itself, can be veryuseful for students reading these notes.

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1 Major Themes

1.1 Grammar of Mathematics

The technical skills of algebra and algebraic manipulation, including using logarithms and trigonometricidentities, are like a grammar to the language of mathematics. Doing mathematics without having theseskills is akin to writing paragraphs and essays without understanding English grammar and spelling. Inthe same way high level conceptual skills in English cannot exist without understanding and masteringthe details of grammar, conceptual skills in mathematics rely on a certain level of technical mastery ofalgebraic techniques.

1.2 Practice

Even though working on technical skills is dry and often requires rote repetition, it is necessary in orderto progress to higher levels of conceptual mathematical learning. To that end, practice is required. Thisdocument, however, is theoretical: it is meant to give a succinct overview of the skills and concepts thatare required. In the interest of relative brevity, there are very few examples and no exercises. Studentsare encouraged to seek other resources for problems and exercises, including online resources and theprecalculus texts in the library.

1.3 Symbol Manipulation

When working through complicated algebra, it is easy to get lost in the symbol-manipulation of techni-cal mathematics. It is very good to remember that all mathematical symbols have meaning and everyalgebraic expression can be translated into a statement about some real quantities. Students shouldavoid moving symbols around while forgetting that the symbols have any meaning. Similarly, much ofalgebra involves solving problems by algorithms, i.e. using a fixed set of steps that consistently leads toa solution. However, students should not be satisfied with just memorizing algorithms – competencyin technical mathematics comes from building intuition that can be applied in unfamiliar settings,not from memorizing recipes for solutions to fixed problems. Being able to apply algebraic techniquesto problems that look unfamiliar is a very important skill. To that end, seek to understand why thealgorithms work instead of just knowing how to step through them.

1.4 Equality

It is amazing how often students forget the basic idea of an equation. Recall what the equals signsmeans: that both sides, though different in form, have the same value. Equations are about balance;anything that changes value on one side must also be done to the other side to preserve balance.However, any operation which only changes form can be done to only one side. Adding a numberobviously changes the value, so it must be done to both sides of an equation. Changing the sum of twofractions into common denominator changes the form but doesn’t change the value, so it can be doneto only one side of the equation.

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2 Sets

While quite abstract, sets form the foundation of modern mathematics. We don’t need a great deal ofset theory for calculus, but we do need a number of basic definitions.

2.1 Basic Definitions

Definition 2.1. A set is a collection of things.

In calculus, the most common sets we see are sets of numbers. If written explicitly, we use this notationto define the set A that contains the four numbers 1, 2, 5 and 10:

A = {1, 2, 5, 10}

Definition 2.2. The things in a set are called elements.

We use this notation to say that 5 is an element of the set A:

5 ∈ A

Sets are not ordered. Two sets are the same if and only if they have the same elements.

{1, 2, 5, 10} = {10, 1, 5, 2}

If we have two sets A and B, there are several basic set operations we can perform:

Definition 2.3. The union of the two sets consists of elements which are in at least one of A or B. Itis written A ∪B.

{1, 3, 7} ∪ {2, 7, 10} = {1, 2, 3, 7, 10}

Definition 2.4. The intersection of the two sets consists of the elements which are in both A and B.It is written A ∩B.

{1, 3, 7} ∩ {2, 7, 10} = {7}

Definition 2.5. The set difference consists of elements which are in A but not in B. It is writtenA\B.

{1, 3, 7}\{2, 7, 10} = {1, 3}

Definition 2.6. The empty set is the unique set with no elements. It is written ∅.

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2.2 Number Sets

There are four important number sets that we use in calculus.

Definition 2.7. The set of natural numbers consists of all positive whole numbers:

N = {1, 2, 3, 4, . . .}

Definition 2.8. The set of intergers consists of all positive and negative whole numbers, includingzero:

Z = {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .}

Definition 2.9. The set of rational numbers consists of all fractions ab where a and b are integers and

b 6= 0. This set is written Q.

Definition 2.10. The set of real numbers consists of all decimal expansions n.a1a2a3a4 . . . where n isan integer and ai are digits between 0 and 9. This set is written R. We often think of R as the numberline and visualize it as such. The real number line has no holes or gaps; this seemlessness property isvery important for calculus.

2.3 Intervals

There are some standard subsets of the real numbers which we must know.

Definition 2.11. A subset of R which is a connected piece of the the number line is called an interval.

If a and b are real numbers with a < b, then we have the following notation for several kinds of intervals.

• The notation [a, b] represents all points on the number line betwee a and b, including the end-points, that is, all x such that a ≤ x ≤ b. This interval is called a closed interval.

• The notation (a, b) represents all points on the number line between a and b, excluding theendpoints, that is, all x such that a < x < b. This interval is called an open interval.

• Likewise, (a, b] is all x such that a < x ≤ b and [a, b) is all x such that a ≤ x < b. These intervalsare called half-open intervals.

Be careful with the notation for open intervals: (a, b) can be notation for an interval on the real numberline, but it can also be notation for coordinates representing a point in R2. It is unfortunate that bothnotations use the same symbols; we have to simply remember from context which notation we areusing.

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3 Fractions

Definition 3.1. A fraction is formed of one mathematical object divided by another. It can be formedfrom any mathematical objects which allow division.

That means we can have fractions with numbers 37 , polynomials x2+1

x3−6x−8 , functions f(x) =√x

ln x or anyother kind of mathematical object which allows divison. Fractions of numbers are the most familiar,but the rules for fractions of numbers nicely extend to other kinds of fractions. Polynomial fractionshave a particular name:

Definition 3.2. A rational expression is a polynomial fraction.

Definition 3.3. The reciprocal of a mathematical object a is the fraction 1a .

The reciprocal of a number a is the fraction 1a . Likewise, the reciprocal of a polynomial p(x) is the

rational expression 1p(x) and the reciprocal of a funciton f(x) is the function 1

f(x) . The reciprocal of

any fraction ab is the inverted fraction b

a .

As soon as we talk about fractions, we must be concerned about division by zero. Zero has no re-ciprocal and any operation involving division by zero is undefined. For numbers, this is relatively

straightforward. If we have f(x)g(x) as a fraction of functions, we must avoid all x values where g(x) = 0.

When we work with fractions, we can multiply numerators and denominators by the same non-zerovalue without changing the fraction. If we have a fraction a

b and c 6= 0 then:

a

b=a

b

c

c=ac

bc

We have to be very careful about c 6= 0 when working with fractions of polynomials or functions.Consider the following equation:

(x+ 1)

(x− 1)=

(x+ 1)(x− 3)

(x− 1)(x− 3)

This is a true statement for all x except x = 3. When x = 3, the factor (x − 3) evaluates to zero, sothe fraction changes from 4

2 = 2 to 00 , which is undefined.

We can also simply or reduce fractions: if we can factor the same term out of the top and the bottomof a fraction, we can remove it.

x2 + 4x+ 3

x2 + 6x+ 9=

(x+ 3)(x+ 1)

(x+ 3)(x+ 3)=x+ 1

x+ 3

The numerator a of fractions can be seperated over the denominators. However, the reverse is not true.Avoid the common mistake of splitting denominators.

a+ b

c=a

c+b

ca

b+ c6= a

b+a

c

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3.1 Common Denonminator

There are two important manipulations of fractions to which we need to attend. The first is addingand subtracting fractions, which is done by common denominator. To move to common denominator,we multiply each fraction by x

x for some appropriate x that constructs the common denominator. Ingeneral:

a

b+c

d=ad

bd+bc

bd=ad+ bc

bda

b− c

d=ad

bd− bc

bd=ad− bcbd

We need to able to find common denominators even when the fractions involve complicated algebraicexpressions. For example:

x2 + 1

x− 1+x2 + 2x+ 1

x− 2=

(x2 + 1)(x− 2)

(x− 1)(x− 2)+

(x− 1)(x2 + 2x+ 1)

(x− 1)(x− 2)

=x3 − 2x2 + x− 2

x2 − 3x+ 2+x3 + x2 − x− 1

x2 − 3x+ 2

=x3 − 2x2 + x− 2 + x3 + x2 − x− 1

x2 − 3x+ 2

=2x3 − x2 − 3

x2 − 3x+ 2

3.2 Nested Fractions

The second important technique with fractions is simplifying nested fractions. The basic idea is thatdividing by a fraction is equivalent to multiplying by its reciprocal. The general case for a fractioncomposed of two fractions is this:

abcd

=a

b· dc

=ad

bc

If the denominator is simply c, we can treat it as c1 .

ab

c=

abc1

=a

bc

Likewise, if the numerator is simply a, we can treat it as a1 .

acd

=a1cd

=ad

c

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4 Exponents

To start, let’s review the basic concept of an exponent.

Definition 4.1. Positive whole exponents are the easiest: they indicate multiplying a number by itselfsome number of times.

24 = 2 · 2 · 2 · 2 = 16

Negative whole exponents indicate the same, but as a reciprocal.

2−3 =1

23=

1

2 · 2 · 2 =1

8

Fractional exponents indicate roots.

813 =

3√

8 = 2

With these three techniques, we can understand the meaning of any rational exponent.

81−54 =

1

8154

=1(

4√

81)5 =

1

35=

1

243

This gives an interpretation for all rational exponents. However, we can also take exponents which are

irrational numbers, such as 2π or 4√7. These are valid mathematical expressions, but their interpre-

tation as exponents is difficult and beyond these notes. In the second term of calculus, we are able togive a proper definition to irrational exponents.

4.1 Laws of Exponents

We would like to know how to work with and manipulate exponents. To that end, we recall some rulesfor working with exponents.

Proposition 4.2. When we multiply terms with the same base, we add exponents.

23 · 24 = 8 · 16 = 256 = 27 = 23+4

xa · xb = xa+b

When we have a negative exponent, we can put it in the denominator as a positive exponent.

2−3 =1

23x−a =

1

xa

7

When we divide terms with the same base, we subtract the exponents.

23

24=

8

16=

1

2= 2−1 = 23−4

xa

xb= xa−b

When we take a product to an exponent, we can distribute the exponent over the product.

(2 · 5)3 = 103 = 1000 = 8 · 125 = 23 · 53(x · y)a = (xa) · (ya)

When we have nested exponents, the order of operations is very important. Without brackets, we assumewe work from the top down. That is:

234

means 2(34) = 281 = 2417851639229258349412352

With exponents bracketed at the base, we multiply the exponents.

(23)4 = 84 = 4096 = 212 = 23·4

(xa)b = xab = (xb)a

In addition to the rules, we have some other useful observations. Be careful to avoid the very commonmistake of distributing exponents over sums and differences:

(x+ y)a 6= xa + ya

(x− y)a 6= xa − ya

Recall that if n is a positive integer, then:

x1n = n

√x

That means that all the rules of exponents apply to roots. Specifically, we have these two identities:

n√xy = ( n

√x)( n√y)

n

√x

y=

n√x

n√y

Since roots are exponents, be careful to avoid the previously mentioned very common mistake ofdistributing over sums and differences:

n√x+ y 6= n

√x+ n√y

n√x− y 6= n

√x− n√y

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Often it is useful to factor expressions with exponents or roots. The general idea is

xa(xb + xc) = xa+b + xa+c

For example, here we can factor x2 out of a square root:√x5 − x7 =

√x4(x− x3) =

√x4√x− x3 = x2

√x− x3

There are a couple of details to point out for square roots. The statement√x = y is equivalent to

y2 = x. In particular, x ≥ 0. Therefore, over the real numbers, we cannot take square roots of negativenumbers.

√−3 is simply undefined. Whenever working with square roots, we must be careful to ensure

that the argument in the square root is always positive.

We could also have observed that√x = y is the same as (−y)2 = x, since the negative sign will

disappear when the term is squared. This leads to an important point: all positive numbers have twosquare roots, one negative and one positive. When we write

√x, we implicitly mean the positive square

root. We write −√x for the negative square root. When we want to refer to both posibilities, we write±√x.

Both of these observations for square roots can apply to any even root. Fourth, sixth, eighths, etc,roots all have positive and negative answers. The notation 4

√x, 6√x and so on always indicates the

positive choice. Even roots can never take negative arguments: 10√−4 is an undefined expression.

For odd roots, there are no problems with multiple answers or negative numbers. (−2)3 = −8 impliesthat 3

√−8 = −2.

5 Logarithms

Definition 5.1. Logarithms are the inverse operation to exponents. There are a couple of ways ofthinking about this inverse operation. First, we may think of translating statements.

log28 = 3 is equivalent to 23 = 8

logab = c is equivalent to ac = b

With this interpretation, we can go from statements about logarithms to statements about exponentsand back again. This is useful for checking results with logarithms. This is also useful for rememberingwhich piece of the logarithmic expressions captures which information: in the previous formula, a isthe base, c is the exponent and b is the result of ac.

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You can also think of logarithms as undoing exponents, so that if we take an exponent and then alogarithm, they cancel each other out.

2(log214) = 14 or log3(312) = 12

a(logab) = b or loga(ab) = b

In either order, one operation cancels out the other. This is very useful for solving equations, wherewe can use logarithms to cancel exponents and vice-versa.

We had four examples in the section on exponents. Here are those four examples translated intologarithms:

24 = 16⇐⇒ log216 = 4

2−3 =1

8⇐⇒ log2

1

8= −3

813 = 2⇐⇒ log8

1

3= 2

81−54 =

1

243⇐⇒ log81

1

243=−5

4

5.1 Manipulations of Logarithms

Much like exponentials, there are useful rules for manipulating logarithms. These are all stated in base2, but they are true in any base.

Proposition 5.2.

When we take the logarithm of a product, we add the logarithms.

log2(8 · 8) = log264 = 6 = 3 + 3 = log28 + log28

log2(x · y) = log2x+ log2y

When we take the logarithm of a quotient, we subtract the logarithms.

log2

(64

8

)= log28 = 3 = 6− 3 = log264− log28

log2

(x

y

)= log2x− log2y

When we take the logarithm of a term with an exponent, we can take the exponent out of the logarithm.This is an extremely useful manipulation.

log2(43) = log264 = 6 = 3 · 2 = 3 log24

log2(xy) = y log2x

10

The proofs of these results can be found by translating into exponents and using the manipulations ofexponents. This highlights how logarihtms and exponents are opposite operations. In particular, youcan think of exponents changing sums and differences into products and quotients, and logarithmschanging products and quotients into sums and differences.

Be aware of the same common mistake as we mentioned for roots and exponents: we can’t distributelogarithms over addition or subtraction:

loga(x+ y) 6= logax+ logay

loga(x− y) 6= logax− logay

5.2 The special base e

There is an irrational number labeled e with the approximate value of e = 2.718281 . . . The formaldefinition of e is given in the first term of calculus. For reasons that will be investigated in the calculusclass, e turns out to be a very good base for exponents and logarithms. The exponential function ex isone of the most important functions in mathematics. The logarithm with base e, loge, is given a specialname and notation: it is called the natural logarithm and written lnx. We can express any logarithmin terms of the natural logarithm by this formula:

logax =lnx

ln a

Here are the rules of logarithms restated for the natural logarithm, as a useful reference.

ln(xy) = lnx+ ln y

ln

(x

y

)= lnx− ln y

ln(xy) = y lnx

ln(x+ y) 6= lnx+ ln y

ln(x− y) 6= lnx− ln y

5.3 Logarithmic Scales

A very common application of logarithms is logarithmic scales, which we can take a few lines to review.The idea behind a logarithmic scale is that instead of measureing the value v of a certain quantity wecare about, we measure logav instead for some suitable base a. Due to our counting system, we oftentake base 10 in logarithmic scales.

Acidity, measured in pH, is a logarithmic scale which measures the concentration of hydrogen ions (H+)in a solution. pH is the negative logarithm of the concentration of hydrogen ions: if the concentration

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is C, then the pH is −log10C. Here is a chart to show how pH and concentration match-up; note thateach step up in pH is one tenth the concentration of the previous step.

pH C: Concentration of H+ in mol/L C as a power of 100 1 100

1 0.1 10−1

2 0.01 10−2

3 0.001 10−3

4 0.0001 10−4

5 0.00001 10−5

6 0.000001 10−6

7 0.0000001 10−7

Other common logarithmic scales include the Richter scale for earthquake intensity and the decibelscale for sound intensity.

6 Inequalities

Sometimes, instead of dealing with equations in algebra, we have to deal with inequalities. These canbe more difficult to manipulate, since the rules are more complicated. It is no longer true that anyoperation we perform on both side of the inequality will preserve the statements. However, we do havesome operations we can still perform. Assuming that a < b, we have these rules:

Proposition 6.1. We can add a number to both sides and preserve the inequality.

a+ c < b+ c

We can subtract a number from both sides and preserve the inequality.

a− c < b− c

We can multiply or divide by a positive number to preserve the inequality.

ac < bc or ac <

bc c > 0

If we multiply or divide by a negative number, the inequality reverses.

ac > bc or ac >

bc c < 0

It we take an odd integer exponent, we preserve the inequality. We can’t say, in general, what happenswith even exponents or non-integer exponents.

a2n+1 < b2n+1

If we take reciprocals, the inequality reverses.

1

a>

1

b

12

For other operations, we often need to work case-by-case to see what each operation does to inequal-ities. Some operations will preserve inequalities nicely, some only partially, and some not at all. Thetechniques of calculus will help us with this problem.

7 Polynomials

Definition 7.1. A polynomial is an expression with positive integer powers of a variable multipliedby constants and added together.

Polynomials in one variable are expressions like 3x3 − 7x2 + 4x − 9 or x34 − 54x16 + 103x9 − 75x4.Sometimes it is useful to have a very general expression: the general form of a polynomial , where x isa variable and the ai are real numbers, is

anxn + an−1x

n−1 + . . .+ a2x2 + a1x

2 + a0

Definition 7.2. The degree of a polynomial is the highest power of the variable.

The polynomial x4 + 5x2 + 19 has degree 4. The polynomial x16 + x24 + 3x19 has degree 24. Note thatthe highest exponent was in the middle of the previous expressions. Since addition is independant oforder, we can write the pieces of a polynomial in any order we wish. Usually we write polynomials withincreasing or decreasing exponents.

Definition 7.3. Polynomials of small degrees are given specific names:

• A polynomial of degree 0 is constant.

• A polynomial of degree 1 is linear.

• A polynomial of degree 2 is quadratic.

• A polynomial of degree 3 is cubic.

• A polynomial of degree 4 is quartic.

• A polynomial of degree 5 is quintic.

Definition 7.4. A root of a polynomial p(x) is a real number a such that when we replace x with ain the polynomial, the resulting value is zero. That is: p(a) = 0.

Roots of a polynomial are not exactly the same as square roots, cube root, etc, even though they usethe same term. The two ideas have a common source, which accounts for the annoying repetition interminology.

13

Polynomials are multiplied together by the distributive law, which is often referred to as expansion.Here are some examples of expansion:

3(x2 + 4x+ 1) = 3x2 + 12x+ 3

(x+ 1)(x− 4) = x(x− 4) + 1(x− 4)

= x2 − 4x+ x− 4 = x2 − 3x− 4

(x2 + 3x+ 7)(2x2 − 4x− 5) = x2(2x2 − 4x− 5) + 3x(2x2 − 4x− 5) + 7(2x2 − 4x− 5)

= (2x4 − 4x3 − 5x2) + (6x3 − 12x2 − 15x) + (14x2 − 28x− 35)

= 2x4 + 2x3 − 3x2 − 43x− 35

(ax+ b)(cx+ d) = ax(cx+ d) + b(cx+ d)

= (ac)x2 + (ad)x+ (bc)x+ (bd)

= (ac)x2 + (ab+ bc)x+ (bd)

The opposite process to expansion is factoring. To factor a polynomial is to express is as the productof small degree polynomials. We’ll investigate factoring and finding roots for quadratics and cubics.

7.1 Quadratics

For a quadratic x2 + ax + b, we look for factors (x + n)(x + m). If we expand (x + n)(x + m) we getx2 + (m+ n)x+mn. Therefore, to match the original polynomial, we look for m and n where nm = band n+m = a. Sometimes (usually when m and n are small integers) we can find m and n by guessing.

x2 + 4x+ 4 = (x+ 2)(x+ 2)

x2 − x− 6 = (x− 3)(x+ 2)

x2 − 9x+ 20 = (x− 4)(x− 5)

x2 − 18x− 595 = (x− 35)(x+ 17)

Notice that in the second factorization x2 − x − 6 = (x − 3)(x + 2), we can see that p(3) = 0 andp(−2) = 0. This is a specific case of a general result.

Proposition 7.5. A polynomial has a linear factor (x−a) if and only if a is a root of the polynomial.In this sense, finding linear factors is the same problem as finding roots.

Not all quadratics factor easily. Equivalently, it is not always this easy to find roots of quadratic.However, we have a general formula.

14

Proposition 7.6. If p(x) = ax2 + bx+ c is a quadratic (so a 6= 0) then the roots of the quadratic aregiven by the quadratic formula, which is this expression:

x =−b±

√b2 − 4ac

2a

This will always give roots if they exist, but there can be problems. The square root term√b2 − 4ac

doesn’t make sense if b2 − 4ac is negative. The quadratic only has roots if this is positive or zero.

Definition 7.7. This expression, b2 − 4ac is called the discriminant of the quadratic.

If the discriminant is negative, there are no roots. If the discriminant is 0, there is a single root. If thediscriminant is positive, there are two roots.

The quadratic formula can be used to factor complicated quadratics, even though it is a bit messy.The quadratic x2 − 6x+ 4 doesn’t factor obviously. The quadratic formula gives roots:

x =6±√

36− 16

2=

6±√

20

2= 3±

√5

The roots are 3+√

5 and 3−√

5. Since roots give us linear factors, the quadratic factors as x2−6x+4 =(x− 3−

√5)(x− 3 +

√5).

There is one special case we can easily recognize for factoring quadratics.

Definition 7.8. A quadratic of the form x2 − a2 is called a difference of squares. It factors easily as(x− a)(x+ a).

Usually we write quadratics as ax2 + bx + c. However, often it is convenient to write them in theso-called vertex form: a(x−m)2 + n.

Definition 7.9. Changing a quadratic from the form ax2 +bx+c into the form a(x−m)2 +n is calledcompleting the square.

The general process goes like this: given ax2 + bx + c, we first write a(x2 + bax) + c. Then to turn

the item in brackets into a perfect square, we add b2

4a . Since we can’t arbitrarily add things, we alsosubtract the same term to balance the expression.

a

(x2 +

b

a

)+b2

4a− b2

4a+ c

Then we take the first new term inside the brackets. Since there is an a that distributes over thebrackets, we must multiply by a in the demoninator to balance.

a

(x2 +

b

a+

b2

4a2

)− b2

4a+ c

Then the item in brackets is (x+ b2a )2, so we have

a

(x+

b

2a

)2

+

(c− b2

4a

)

15

7.2 Cubics

Factoring becomes a more difficult problem as the degree increases. Factoring cubics is already a muchmore difficult problem than factoring quadratics. There is a formula for cubics, like the quadraticformula, but it is much more complicated and involves using

√−1 and the complex number system

which allows for this operation.

However, there are a couple things we can do for special cases. Sometimes, we can look for one rootsimply by inspection. Take, for example, the polynomial x3 + 3x2− 2x− 2. Just by trying numbers, wecan discover that x = 1 is a root of the polynomial. Therefore (x−1) is a factor, so x3 +3x2−2x−2 =(x−1)(ax2 +bx+c). We can calculate a, b, and c by multiplying the two factors together and matchingwith the original polynomial. That results in:

x3 + 3x2 − 2x− 2 = (x− 1)(x2 + 4x+ 2)

If we wanted to factor further, we would look to factor the quadratic. This quadratic doesn’t factornicely, but the quadratic formula gives roots of −2±

√2. Therefore

x3 + 3x2 − 2x− 2 = (x− 1)(x2 + 4x+ 2) = (x− 1)(x+ 2−√

2)(x− 2−√

2)

In general, if we can’t guess a specific root, finding this decomposition is very difficult. However, thereare two special forms of cubics which are easy to factor.

Definition 7.10. A cubic of the form x3 + a3 is called a sum of cubes. A cubic of the form x3 − a3 iscalled a difference of cubes.

These factor as:

x3 − a3 = (x− a)(x2 + ax+ a2)

x3 + a3 = (x+ a)(x2 − ax+ a2)

In both cases, they can’t be factored any further. The discriminant of the remaining quadratic isa2 − 2a2, which is always negative.

16

8 Trigonometry

There are two basic trigonometric functions: sine and cosine. In high school these are usually definedby triangles. However, in higher mathematics, we usually define sine and cosine using the circle. Weuse the Greek letter theta, written θ, as the angle. We measure θ ins radians.

Definition 8.1. If (x, y) is a point on the circle of radius 1 and θ is the angle between the positive xaxis and the radius to the point (x, y), then x = cos θ and y = sin θ.

(x, y)

x = cos θ

y = sin θθ

The circle of radius 1 is described by the equation x2 +y2 = 1 in R2. Since x = cos θ and y = sin θ, thisgives the identity cos2 θ + sin2 θ = 1, which is the most important identity in trigonometry. Thoughless directly, other trigonometric identities also come from the geometric definition.

Definition 8.2. The other four trigonometric functions are defined in terms of sine and cosine:

Tangent: tan θ =sin θ

cos θ

Cotangent: cot θ =cos θ

sin θ

Secant: sec θ =1

cos θ

Cosecant: csc θ =1

sin θ

8.1 Radians

While degrees are used for measuring angles at the high-school level, in higher mathematics, we measureangles almost exclusively in radians. 2π radians is a full circle, π radians is a half circle and π/2 radiusis a quarter circle.

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8.2 Graphs of Trigonometric Functions

Sine and cosine have familiar graphs:

y

xcosx

sin x

Definition 8.3. In this graph, the height of the wave is called amplitude. The standard functionf(x) = sinx has amplitude 1. f(x) = 3 sinx has amplitude 3. The distance between two peaks of thewave is called the period of wavelength. The standard function f(x) = sinx has period 2π. (The useof radians is implicit in this). The function f(x) = sin(4x) has period π/2. The function f(x) = sin x

4has period 8π. The reciprocal of period is called frequency, so f(x) = sinx has frequencey 1

2π .

The other four trigonometric functions have denominators which are potentially zero, so their graphswill have asymptotes. An asymptote is a vertical or horizontal line which the graph of a functionapproaches. The graph of tangent should be familiar, with its many vertical asymptotes:

y

x

tan x

8.3 Special Triangles

There are two very important triangles, with their associated trigonometric identities, that frequentlycome up in mathematics. These are called special triangles. The first has angles of π/2, π/4 and π/4.

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The second has angles of π/2, π/3 and π/6. The following table records all the values of trigonometricfunctions for angles in special triangles.

x sinx cosx tanx

0 0 1 0

π6

12

√32

√33

π4

√22

√22 1

π3

√32

12

√3

π2 1 0 undefined

8.4 Trigonometric Identities

We have already mentioned the most important identity: sin2 x + cos2 x = 1. There are many more.The following list is also found in the reference documents:

Squares Shifts

sin2 x+ cos2 x = 1 sin(x+ π

2

)= cosx

1 + tan2 x = sec2 x cos(x− π

2

)= sinx

1 + cot2 x = csc2 x tan(π2 − x

)= cotx

Additions Subtractions

sin(x+ y) = sinx cos y + cosx sin y sin(x− y) = sinx cos y − cosx sin y

cos(x+ y) = cosx cos y − sinx sin y cos(x− y) = cosx cos y + sinx sin y

tan(x+ y) = tan x+tan y1−tan x tan y tan(x− y) = tan x−tan y

1+tan x tan y

Double Angles Half Angles

sin 2x = 2 sinx cosx sin2 x = 1−cos 2x2

cos 2x = cos2 x− sin2 x cos2 x = 1+cos 2x2

tan 2x = 2 tan x1−tan2 x

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Some of these identities relate to the process of solving triangles. By solving a triangle, we mean findingunknown angles and side lengths once we know some of the angles and side lengths. Solving trianglesis an important part of trigonometry. There are two special identities which are particularly useful forsolving triangles: the Sine Law and the Cosine Law. If a triangle has side lengths a, b and c with anglesA, B and C such that A is opposite the side a, B is opposite the side b and C is opposite the side c,then the two laws are:

sinA

a=

sinB

b=

sinC

cSine Law

c2 = a2 + b2 − 2ab cosC Cosine Law

As with exponents, roots and logarithms, avoid the common mistake of distributing trigonometricfunctions over sums.

sin(x+ y) 6= sinx+ sin y

cos(x+ y) 6= cosx+ cos y

tan(x+ y) 6= tanx+ tan y

8.5 Inverse Trigonometric Functions

Once we have a process in mathematics, we often ask how to undo the process, i.e., how to go backwards.If f(x) = x2, undoing the squaring process is accomplished by the square root: f−1(x) =

√x. The

logarithm is undoing the exponential, so if f(x) = 2x then f−1(x) = log2x. A later section of thisdocument discusses inverting functions in general.

We want to invert the trigonometric functions, to find their opposite operations. However, we need torestrict the domains of the trigonometric functions in order to do this.

For f(x) = sinx, the standard restriction of the domain is [−π/2, π/2]. Then sinx : [−π/2, π/2] →[−1, 1]. The inverse function is arcsin(x) : [−1, 1] → [−π, π]. This is sometimes written sin−1(x),particularly on calculators.

This is an opposite operation, so for any x ∈ [−π/2, π/2] we have arcsin(sin(x)) = x and for anyx ∈ [−1, 1] we have sin(arcsin(x)) = x. In expressions and equations, we use the inverse functionsto cancel off the trigonometric functions. Alternatively, we can simply see the inverse trigonometricfunction as this equivalence:

sin a = b is equivalent to a = arcsin b

For f(x) = cosx, the standard restriction of the domain is [0, π], so cosx : [0, π]→ [−1, 1]. The inversefunction is arccos(x) : [−1, 1]→ [0, π]. This is sometimes written cos−1(x), particularly on calculators.

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For f(x) = tanx, the standard restriction of the domain is (−π/2, π/2) so that tanx : (−π/2, π/2)→R. Then the inverse function is arctanx : R → (−π/2, π/2). This is sometimes written tan−1(x),particularly on calculators.

The remaining three trigonometric function also have standard restrictions and inverses, which aresummarized in the following table.

function domain range inverse

sinx[−π

2 ,π2

][−1, 1] arcsinx

cosx [0, π] [−1, 1] arccosx

tanx(−π

2 ,π2

)R arctanx

secx[0, π2

)[1,∞) arcsec x

cscx(0, π2

][1,∞) arccsc x

cotx (0, π) R arccot x

9 Cartesian Geometry

We need to be familiar with the Cartesian plane and its geometry, particularly the geometry of lines.

Definition 9.1. The Cartesian plane is written R2 and consists of ordered pairs (x, y) of real numbers.

It has two axes: the x axis drawn horizontally and increasing to the right, and the y axis drawnvertically and increasing upwards. The origin is the point (0, 0) located at the intersection of the twoaxes.

Equations in x and y correspond to shapes in the plane. The most common of these are lines. Weshould be familiar with two forms of equations of lines. If we gather all the variables together, we getthe general form of the equation of line: ax+ by+ c = 0. However, this isn’t geometrically that useful.As long as the line is not vertical, we can change this into slope-intercept form y = mx+ b where m isthe slope of the line and b in the y value of the intercept with the y axis.

This value m in the slope-intercept form y = mx + b measures the slope of the line, which is animportant property. The general definition is this:

Definition 9.2. If (x1, y1) and (x2, y2) are two points on a line, the slope of the line is given by thefraction y2−y1

x2−x1. Vertical lines have an undefined slope.

This section on Cartesian geometry is very brief. Please consult my notes on vector geometry for moredetails.

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y

x

(3, 6)

(0, 0)

(4,−4)

(−5,−2)

(−7, 7)

10 Functions

Though this is the last section of these notes, it may be the most important. We could define calculusas the study of the behaviour of functions. Therefore, having a solid conceptual grasp of functions isnecessary for proceeding into calculus.

Definition 10.1. If A and B are sets, then a function f : A→ B is a rule which takes elements a ∈ Aand assigns elements f(a) ∈ B. We think of functions as active agents: they act on elements of the setA and change them into elements of B. For calculus and many other parts of mathematics, we assumethat A and B are sets of real numbers.

While defining a function as a rule is a clean and elegant definition, the idea of a function is a rich andcomplex one. Many mathematicians and educators have observed that there are (at least) four majorideas associated with functions:

• A function is a rule stating how elements of one set are assigned to elements of a second set. Thisis the definition we started with.

• A function is a machine, which takes input (things in the first set) and alters the input to produceoutput (things in the second set). In this way we think of f(x) = x2 as the machine that squaresnumbers.

• A function is a way of encoding relationships and dependencies between quantities. If t is timeand d is distance, then d = f(t) is a way of saying that distance is related to and dependent ontime.

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• A function, geometrically, is a graph. A graph is a curved path through R2 which satisfies avertical line test: any vertical line only meets the path at most once. Any such path represents afunction by identifying its points with (x, f(x)), i.e. the x coordinate of the point is the input ofthe function and the y coordinate of the point is the output of the function.

10.1 Operations on Functions

If f and g are two functions A→ B and n ∈ N, then there are a number of operations we can do withf and g. Pointwise operations are arithmetic and algebraic operations that we evaluate seperately ateach value of the function.

(f + g)(x) = f(x) + g(x)

(f − g)(x) = f(x)− g(x)

(fg)(x) = f(x)g(x)

f

g(x) =

f(x)

g(x)

(fg)(x) = (f(x))g(x)

(fn)(x) = (f(x))n

( n√

(f))(x) = n√f(x)

In addition to pointwise operations, the most important operation on functions is composition.

Definition 10.2. If f : A→ B and g : B → C are two functions, then the composition of f and g iswritten g ◦ f and defined as

(g ◦ f)(x) = g(f(x))

Note that the output f has to be valid input for g, since the functions happen one after the other. Alsonote that we write composition from right to left: in g ◦ f , the function f acts first and the functiong acts second. This notation make sense if we think of f and g acting on a variable x and the agentcloser to x gets to act first.

It is important to recognize the difference between pointwise operations and composition. For examplef(x) = (sinx) · (x2) is the (pointwise) multiplication of two functions: sinx and x2. However, f(x) =sin(x2) is the composition of sinx and x2.

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10.2 Graphs of Functions

We use graphs as the main tool for visualizing functions.

Definition 10.3. If f : A → R is a function where A is a subset of real numbers, then charting thepoints (x, f(x)) in the Cartesian plane gives the graph of the function.

Definition 10.4. An asymptote is a straight line which the graph of a function approaches. The graphmust get arbitrarily close to the line, but never touch it. If the line is a vertical line, we call it a verticalasymptote. Likewise, if the line is a horizontal line, we call it a horizontal asymptote.

The first figure on the next page contains the graphs of three common functions. The second figure isthe graph of a function with two horizontal asymptotes. For graphs of many common functions, referto the library of functions document.

y

x

g(x) = x2

f(x) = ex

h(x) = −3x+ 1

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y

x

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10.3 Domain and Range

Definition 10.5. The domain of a function is the set of all inputs in the definition of the function.

If f : A → B is a function, its domain is A. If a function is just given by a formula, say f(x) =x2 −

√x− 3, then the domain is assumed to be all real numbers where the function is defined, in this

case x ≥ 3.

For most purposes, there are three domain restrictions we have to worry about:

• We can’t divide by zero. Any inputs that would involve division by zero cannot be in the domain.

• We can’t take even roots of negative numbers. Any inputs that would involve even roots ofnegative numbers cannot be in the domain. (Odd roots are fine: 3

√−27 = −3, for example, is

perfectly well defined.)

• We can’t take logarithms of negative numbers or zero. Any inputs that would involve logarithmsof negative numbers or zero cannot be in the domain.

As we move on in mathematics, there can be other restrictions on the domain for specific functions.For example, the inverse trigonometric function f(x) = arcsin(x) has domain [−π, π]. This restrictionmust be respected whenever we use arcsine in more complicated compound functions.

We can also intentionally restrict the domain of a function. The function f(x) = x2 is defined for allreal numbers, but we could consider f(x) = x2 only acting on [0,∞), if we wished to do so. There area number of good reasons in mathematic to restrict domains to smaller sets.

Definition 10.6. The range of a function is the set of all outputs of the function.

In general, range is much more difficult to figure out than domain. For domain, we mostly have to makesure we avoid certain undefined operations, such as division by zero. For range, we actually have towork out what the function is doing and construct an idea of all possible outputs. There is no generalprocess or formula for doing this.

For some functions, we can figure it out logically. The function f(x) = x2 outputs squares, which arealways positive. However, they can be very close to zero, if x is a small number. Also 02 = 0, so zerois in the output. This lets us argue that the range is [0,∞).

If we think about the sine function, using the definition as the y coordinate of points around the unitcircle, we can observe that the maximum of that y coordinate if 1 and the minimum is −1. Thereforethe range of sine is [−1, 1]. Likewise for cosine. However, we can think of the tangent function as theratio of the two coordinates y/x, which is also the slope of the radius connecting the centre to thepoint (x, y). This slope can be arbitraily steep, both in a positive and negative slopes. Therefore, wecan conclude that the range of the tangent function is R.

You can see how, in all these cases, there is a logical argument that leads us to range. These argumentscan be simple and succinct or very long and complicated. We said it before, but it bears repeating thatthere is no uniform formula or process for calculating range.

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10.4 Properties of Functions

Definition 10.7. A function f of real numbers is called bounded above if there is a number M ∈ Rsuch that f(x) ≤ M for all x in the domain of f . A function is called bounded below if there is anumber N ∈ R such that f(x) ≥ N for all x in the domain of f . If a function is both bounded aboveand below, we say is is bounded.

We could have stated this in terms of range as well. A function is bounded above if its range has anupper limit; a function is bounded below if its range has a lower limit.

Definition 10.8. A function f of real numbers is called increasing if for all x, y in the domain of f ,x < y implies f(x) < y. Similarly, f is called decreasing if the implication is f(x) > f(y) A function iscalled monotonic if it is either increasing or decreasing.

Increasing and decreasing are easy to read from the graph of a function: increasing means the graph isgoing up as we move from left to right, and decreasing means the graph is going down. Monotonic isa strange term, but it is convenient later when we speak about inverting functions.

When we say a function is increasing or decreasing, we mean that the implication applies for thewhole domain: the function is always going up or always going down. When we have a function whichchanges direction, sometimes going up and sometimes going down (such as the sine function) we don’tsay the function is both decreasing and increasing; instead we say the function is neither decreasingnor increasing.

There are three properties of functions which capture the symmetry of a function.

Definition 10.9. A function is even if for all x in the domain, f(−x) = f(x). A function is odd if forall x in the domain, f(−x) = −f(x).

Even and odd are useful symmetries. In terms of the graph of a function, an even symmetry is amirror symmetry over the y axis and an odd symmetry is a rotational symmetry by π radians (180degrees) around the origin. Most functions are neither even or odd. Some important even functions aref(x) = x2 and f(x) = cosx. Some important odd functions are f(x) = x3 and f(x) = sinx.

Definition 10.10. A function is periodic if there is a number a, called the period, such that for all xin the domain of the function, f(x+ a) = f(x).

The most common periodic functions are the trigonometric functions: sin, cos and tan are all periodicwith period 2π.

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10.5 Classes of Function

There are many different types and classes of functions. We’ll define some of the familiar ones here.Each class in this list includes all previous classes.

Definition 10.11. A function which has only one output, regardless of the input, is called a constantfunction. Examples are f(x) = 7 and f(x) = −π.

Definition 10.12. If a and b are real numbers, then a function of the form f(x) = ax+ b is called alinear function. The graph of a linear function is a straight line. (Constant functions are included here,since a can be zero).

Definition 10.13. If a function has this form: f(x) = anxn + an−1x

n−1 + . . .+ a2x2 + a1x+ a0, it is

called a polynomial function. The degree of the polynomial is also called the degree of the function. Apolynomial function of degree 0 is a constant function. A polynomial function of degree 1 is a linearfunction. A polynomial function of degree 2 is a quadradic, degree 3 a cubic, degree 4 a quartic anddegree 5 a quintic.

Definition 10.14. A function of the form f(x) = p(x)q(x) where p and q are polynomials is called a

rational function. This includes polynomial functions, since we can take the case where q(x) = 1.

Definition 10.15. Any function which is formed of some combination of variables with addition,subtraction, multiplication, division, and rational exponents (including roots) is called an algebraicfunction.

Definition 10.16. Any function that combines an algebraic function with trigonometric functions,inverse trigonometric functions, exponentials or logarithms is called a transcendental function.

10.6 Piecewise Functions

Sometimes, a function uses different definitions for different pieces of its domain. We call these functionspiecewise functions, and they are written with a one-sided large bracket notation. For example, afunction which uses the definition f(x) = sinx for negative x and f(x) = x2 for positive x and 0,would be written:

f(x) =

{sinx x < 0x2 x ≥ 0

Piecewise definitions are very useful for a number of applications. It is quite common in mathematicsto have a sudden change in the behaviour of a function, so that we need different formulae for differentparts of the domain. A common example is the absolute value function f(x) = |x|. If x is positive or 0,this is just x. If x is negative, we multiply by −1 to make it positive. In terms of a piecewise function,this is:

|x| ={−x x < 0x x ≥ 0

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When working with absolute value, or piecewise function in general, it is necessary to work in cases.For example, let’s say we had an expression involving |x2 − 4|. When x2 − 4 is positive, we leave thisalone, but when it is negative, we change it to 4 − x2. We can’t do both at the same time, so we usecases. x2 − 4 is negative when x ∈ (−2, 2), so those x would form one case. x2 − 4 is zero or positivewhen x ≥ 2 or x ≤ −2. Those would form the remaining cases.

10.7 Inverse Functions

If we have a function f : A → B, we often want to ask how we can undo what the function does. Isthere another function or another process which reverses f , i.e., sends f(x) back to the x it came from?For some functions (but not all), we can find such a reversal.

Definition 10.17. If f : A → B is a function, then its inverse function is a function f−1 : B → Awhich undoes the action of f . That is f(f−1(x)) = x and f−1(f(x)) = x.

We already seen some examples. The inverse of f(x) = x2 is f−1(x) =√x. The inverse of f(x) = ex is

f−1(x) = lnx. We defined the trigonometric inverses arcsin, arccos and so on. However, we still havea general question: when can we be sure an inverse exists?

Proposition 10.18. If f is a monotonic function, then f−1 exists.

In general, if f is not monotonic, we will restrict its domain until it is monotonic and we can define aninverse. This is why we restricted the domains of the trigonometric functions.

To try to find an inverse function, we take the equation y = f(x) and try to solve for x. If possible,the resulting expression will be the formula for the inverse function. For example, take the functionf(x) =

√x3 − 8 for x ≥ 2.

f(x) =√x3 − 8

y =√x3 − 8

y2 = x3 − 8

y2 + 8 = x3

3√y2 + 8 = x

f−1(x) =3√x2 + 8

Finally, an inverse function will switch domain and range: The domain of f is the range of f−1 andthe range of f is the domain of f−1.

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