Pre--Calculus Review, Study notes of Mathematics

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Precalculus Review

Ethan D. Bloch

Revised draft

August 13, 2020

+ Do not circulate or post

4 CONTENTS

1.1 (^) Algebra

Calculus makes use of precalculus—hence the name of the latter—but to do precalculus, a solid knowledge of basic algebra is needed. We review here a few of the most important ideas from algebra that are needed for calculus.

Types of Numbers

Precalculus, and calculus, takes place within the context of the real numbers. Within the real numbers, there are some import special types of numbers that are frequently used in mathematics.

Types of Numbers

1. The real numbers, denoted R, are all the numbers on the number line, including positive num- bers, negative numbers, zero, whole numbers, fractions, and all other numbers (such as

and π ).

2. The rational numbers, denoted Q, are all numbers that are expressible as fractions, for example 2 3 or^ −^0_._ 5.
3. The integers, denoted Z, are the numbers − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 ,....
4. The natural numbers, also called the positive integers, denoted N, are the numbers 1 , 2 , 3 , 4 ,....

Note that all natural numbers are integers, and all integers are rational numbers, and all rational num- bers are real numbers, but not the other way around. A collection of numbers that is even larger than the set of real numbers is the set of complex numbers, denoted C. It is not assumed that the reader is familiar with the complex numbers. These numbers are not used in Calculus I and Calculus II ; they do arise in Introduction to Linear Algebra and Ordinary Differential Equations , and they will be discussed there.

We will, at times, be using the symbols ∞ and −∞ to denote “infinity” and “negative infinity,” respectively. These words are written in quotes to emphasize the following.

• Error Warning The symbols ∞ and −∞ are not numbers. These symbols represent what hap- pens as we take numbers that get larger and larger without bound (going to ∞) and get smaller and smaller (meaning negative numbers having larger and larger magnitude).

For example, the numbers 2 , 4 , 8 , 16 , 32 ,... are “going to ∞,” and the numbers − 1 , − 3 , − 5 , − 7 , − 9 ,... are “going to −∞.”

1.1. ALGEBRA 5

• Error Warning Do not try to use the symbols ∞ and −∞ in algebraic expressions (for example “∞ + 5”).

Intervals

Intervals are a very useful type of collections of real numbers. An interval is the set of all numbers between two fixed numbers, where the endpoints might or might not be included in the interval. The different types of interval are as follows.

Intervals

Let a and b be real numbers. Suppose that a ≤ b.

Notation Type of Interval Definition ( a, b ) open bounded interval a < x < b [ a, b ] closed bounded interval a ≤ x ≤ b [ a, b ) half-open interval a ≤ x < b ( a, b ] half-open interval a < x ≤ b ( a, ∞) open unbounded interval a < x (−∞ , b ) open unbounded interval x < b (−∞ , ∞) open unbounded interval all real numbers [ a, ∞) closed unbounded interval a ≤ x (−∞ , b ] closed unbounded interval x ≤ b

For example, the interval [2 , 5] is the set of all real numbers x such that 2 ≤ x ≤ 5. The interval (3 , ∞) is the set of all real numbers x such that 3 < x.

• Error Warning The notation ( a, b ), for example (1 , 6), is used to mean different things in math- ematics. In the present context the notation (1 , 6) means the interval from 1 to 6, not including the endpoints. On the other hand, when discussing points in the plane (usually denoted R^2 ), the no- tation (1 , 6) means the point in R^2 with x -coordinate 1 and y -coordinate 6. The fact that the same mathematical notation can mean very different things in different contexts can be confusing, but it is a historical accident with which we are now stuck. Fortunately, the meaning of notation such as ( a, b ) can usually be figured out from the context.

• Error Warning The symbols ∞ and −∞ are not numbers, and cannot be included in an interval. Hence, there is no interval of the form “[2 , ∞].”

Absolute Value

A very useful function for working with numbers is the absolute value function, which is defined as follows.

1.1. ALGEBRA 7

Solving Quadratic Equations

There are two methods to solve the equation x^2 + bx + c = 0.

1. If numbers r and s can be found such that r + s = b and rs = c , then x^2 + bx + c = ( x + r )( x + s ), and the roots of x^2 + bx + c = 0 are x = − r and x = − s.
2. The roots of ax^2 + bx + c = 0 are x =

− b ±

b^2 − 4 ac 2 a

, provided b^2 − 4 ac ≥ 0.

Fractions and Rational Expressions

A elementary topic that is needed for calculus, and that, for whatever reason, is something that not every student of calculus knows sufficiently well, is the addition, subtraction, multiplication and division of fractions. Specifically, for calculus we need to add, subtract, multiply and divide fractions that involve letters as well as numbers, and fractions that have fractions in their numerators and denominators. One of the key idea to keep in mind in algebra is that letters in algebra simply stand for numbers that we don’t know their values, and we therefore treat letters exactly the same as we would treat numbers. In particular, the familiar rules for adding, subtracting, multiplying and dividing fractions with numbers work just as well for fractions with letters, and for built-up fractions. One other thing to keep in mind is that when dealing with built-up fractions, which have fractions in their numerators and/or denominators, it is important to distinguish the main fraction line from the subsidiary fraction lines. Visually, the best way to make this distinction is to write the main fraction line longer than the other fraction lines. Even better, the main fraction line should be written not only longer than the other fraction lines, but should be written level with the equals sign. There are three particular types of built-up fractions that can cause confusion, and which we examine. The way to simplify these types of fractions should not be memorized. Rather, all such fractions should be simplified using the basic rules for adding, subtracting, multiplying and dividing fractions.

1. Simplify

a b c

This fraction can be simplified by rewriting the denominator as c 1 , yielding a b c

a b c 1

a b

c

a bc

2. Simplify

a b c

This fraction can be simplified by rewriting the numerator as 1 a , yielding

a b c

a 1 b c

a 1 ·^

c b =^

ac b.

3. Simplify

a b +^

c d e

This fraction can be simplified by first adding the two fractions in the numerator, and then using the method of Item 1, yielding a b +^

c d e =

ad + bc bd e =

ad + bc bd e 1

ad + bc bd ·^

e =^

ad + bc bde.

8 CONTENTS

The following two examples are both used in calculus.

Example 1

Simplify

1 x + h −^

1 x h

Solution We compute

1 x + h −^

1 x h

x −( x + h ) x ( x + h ) h

− h x ( x + h ) h 1

− h x ( x + h )

h

x ( x + h )

Example 2

Simplify

x + h −

x h

Solution Here we use a little trick, which is

x + h −

x h

x + h −

x h

x + h +

x √ x + h +

x

= ( x^ +^ h )^ −^ x h (

x + h +

x )

=

h h (

x + h +

x )

x + h +

x

The expression

x + h +

x , which is used in order to remove the square roots in the numerator, is referred to as the “conjugate” of

x + h −

x.

EXERCISES

1–4 Multiply and then simplify each expression.

1. (3 x + 5)( x^2 − 2 x + 4)
2. (2 m + 3 n )(3 m^2 + 5 mn − n^2 )
3. (2 y + 1)( y − 5)(3 y + 4)
4. 5 p^2 ( p + 3)(3 p^2 + 4)

5–8 Multiply and then simplify each expression by using basic formulas.

5. (5 a + 3)^2 6. (3 m − 4 n )^2
6. (5 y + 1)(5 y − 1) 8. (2 s^2 − t )(2 s^2 + t )

10 CONTENTS

1.2 (^) Functions and Graphs

Functions are the main ingredient in calculus. The two main things we do in calculus, namely, derivatives and integrals, and things that are done to functions. Functions are also a unifying approach in mathematics. For example, whereas logarithms and trigonom- etry seem to be very different, what we are interested in here is logarithmic functions and trigonometric functions, and, even though these two types of functions arise from very different considerations, as func- tions we treat them just as we do any other functions. One thing to keep in mind about functions is that it is not correct to think of functions simply as formulas, for example f ( x ) = x^2. Whereas it is true that many useful functions are given by formulas, there are also useful functions that are not given by single formulas, not to mention functions not given by formulas at all. The most basic idea of a function is that it takes some sort of object as input (in calculus the input is numbers or vectors, though other types of input are used elsewhere), and for each possible input, there is one and only one output. There are different ways of representing functions, including

1. Verbally
2. Numerically (table of values)
3. Graphically
4. By formula (some says “algebraically,” but that isn’t correct).

All these methods of describing a function are equivalent, and it is important to be able to go from one method to the other, for example to go from formula to graph and vice-versa.

Domain of a Function

Every function can take certain things as inputs. For example, the function f ( x ) defined by the formula f ( x ) = x^2 can take all real numbers as inputs, whereas the function g ( x ) defined by the formula g ( x ) = ln x can take only positive real numbers as inputs. In our present context, we are considering functions with real numbers as inputs, and we then have the following concepts.

Domain and Range

Let f ( x ) be a function with real numbers as inputs.

1. The domain of f ( x ) is the set of all possible real numbers for which the function produces an output.
2. The range of f ( x ) is the set of all outputs of the function, when everything in the domain of f ( x ) is substituted into the function.

The range of a function can be useful in some contexts, though for our purpose the domain is the much more important concept. In more advanced mathematics, the concept of the domain of a function, which takes on even more importance, is slightly more general than we are using here.

1.2. FUNCTIONS AND GRAPHS 11

There is no definitive method for finding the domain of a function. However, there are a few things to keep in mind. For example, because it is not possible to divide by zero, we exclude anything from the domain that would lead to dividing by zero. Hence, the domain of the function defined by the formula

f ( x ) = 1 x − 2

is the set of all real numbers other than 2. Other standard considerations when finding the domain of a function is that we cannot take the square root of a negative number (we are considering only real numbers here); we cannot take the logarithms of a

negative number or zero; and we cannot take the tangent of

π 2

3 π 2

, etc.

Substituting Numbers and Expressions into Functions

The point of a function is that we put things into it, and get something out of it for each thing we put into it. For example, let f ( x ) be the function defined by the formula f ( x ) = x^2. Clearly, if we put 3 into the function, we get f (3) = 9 as the output. It can certainly happen that different inputs produce the same output. For example, we note that f (−3) = 9 for this function. The crucial thing to observe is that a single input produces a single output. For example, let g ( x ) be the function defined by the formula g ( x ) =

x. First, we observe that the domain of g ( x ) is the set of all non-negative numbers. More importantly, we note that when we write

x , we mean only the positive square root of x. For example, we have g (4) = 2. It is certainly true that −2 is also a square root of 4, but we cannot say that g (4) is ±2, because that would give us two outputs for the single input 4. Hence, we use the standard convention that

x always means the positive square root of x. If we want to obtain the negative square root of a number, we would need a different function, namely, the function h ( x ) defined by the formula h ( x ) = −

x. For calculus, we need to substitute not only single numbers into functions, but also more complicated expression. For example, let f ( x ) be the function defined by the formula f ( x ) = x^2. Then we will need to compute the expression f ( x + h ), which is given by

f ( x + h ) = ( x + h )^2 = x^2 + 2 xh + h^2_._

• Error Warning In the above example, it is important to recognize that f ( x + h ) is not the same as f ( x ) + h , which would be x^2 + h. That is, when we add h to x “inside” the function, that is not the same as adding h to the “outside” of the function.

Graphs of Functions

Just as a person can be encountered in different ways (for example, in person, by phone, by email, on social media), so too can a function be seen in different ways. One way of describing a function is via a formula, for example the function f ( x ) defined by the formula f ( x ) = x^2. Another way of describing the same function is visually, via its graph. The graph of a function of the form y = f ( x ) is the subset of the plane consisting of all points ( a, b ) that satisfy the equation f ( a ) = b. For example, for the function f ( x ) defined by the formula f ( x ) = x^2 , the point (3 , 9) is in the graph of the function, because f (3) = 3^2 = 9. To find all the points on the graph of a function, the most direct method would be to take every number in the domain of the function and put it into the function to find the output, and then plot all the points obtained in this way. Of course, doing that is not physically possible, because are infinitely many real numbers. Nonetheless, we can figure out what the graphs of many functions looks like. For example, the graph of the function f ( x ) defined by the formula f ( x ) = x^2 is seen in Figure 1 of this section.

1.2. FUNCTIONS AND GRAPHS 13

Basic Graphs

1. y = c 2. y = x
2. y = − x 4. y = | x |

New Graphs from Old

One of the main methods of graphing functions is to do so by modifying the graphs of familiar functions. There are a number of such modifications, which are summarized as follows.

New Graphs from Old

Let f ( x ) be a function, and let c be a positive real number.

14 CONTENTS

Function Type of Modification y = f ( x + c ) shift the graph of y = f ( x ) to the left by c units y = f ( x − c ) shift the graph of y = f ( x ) to the right by c units y = f ( x ) + c shift the graph of y = f ( x ) upward by c units y = f ( x ) − c shift the graph of y = f ( x ) downward by c units y = cf ( x ) stretch the graph of y = f ( x ) vertically by a factor of c y = f ( cx ) stretch the graph of y = f ( x ) horizontally by a factor of c y = − f ( x ) reflect the graph of y = f ( x ) in the x -axis y = f (− x ) reflect the graph of y = f ( x ) in the y -axis

Of course, the various types of modifications listed above can be combined.

Example 1

Sketch the graph of y = − 2 | x − 1 | + 3.

Solution As seen in the following figures, we sketch this graph in steps, starting with the graph

of y = | x |, and then doing one modification of the graph at a time until we obtain the desired result.

1. y = | x | 2. y = | x − 1 |
2. y = 2| x − 1 | 4. y = − 2 | x − 1 |

16 CONTENTS

9–12 Sketch the graph of each function, where the graph of y = f ( x ) is seen below.

9. y = f ( x − 3) 10. y = f ( x ) + 2
10. y = − 2 f ( x ) 12. y = 2 f ( x + 1) − 3

13–16 Sketch the graph of each function.

13. y = | x + 2| 14. y = | x | + 2
14. y = 2| x − 3 | 16. y = −| x | + 1

17–20 Sketch the graph of each function.

17. y =

  

x^2 + 1 , if x ≥ 0 − x, if x < 0.

18. y =

  

| x − 3 | , if x ≥ 1 5 x^2 , if x < 1.

19. y =

    

3 , if x ≥ 1 x, if − 1 ≤ x < 1 − 3 , if x < −1.

20. y =

    

sin x, if x ≥ π 2 tan x, if − π 2 ≤ x < π 2 cos x, if x < − π 2.

1.3. LINEAR FUNCTIONS 17

1.3 (^) Linear Functions

Linear functions appear throughout mathematics and the application of mathematics in the sciences and social sciences. In particular, linear functions play a crucial role in calculus, because the derivative of a function is just the slope of the tangent line at each point of the graph of the function. Linear functions are functions whose graphs are straight lines. It is assumed that you are familiar with straight lines from a geometric point of view. Whereas straight lines exist in both the plane and three-dimensional space (and higher-dimensional space as well), at present we are concerned only with lines in the plane. It is very important to distinguish between all lines in the plane on the one hand, and lines that are graphs of functions on the other hand. The difference between these two types of lines is that lines that are graphs of functions cannot be vertical.

Slope and y-intercept

A straight line in the plane that is the graph of a function has two significant numbers associated with it, namely, the slope and the y -intercept, which are defined as follows.

Slope and y -intercept

1. Let ( x 0 , y 0 ) and ( x 1 , y 1 ) be points. Suppose x 0 , x 1. The slope of the line containing ( x 0 , y 0 ) and ( x 1 , y 1 ) is m =

y 1 − y 0 x 1 − x 0

2. The y -intercept of a line is the value of y where the line intersects the y -axis.

The important thing to observe about the definition of the slope of a line is that no matter which two

points on the line ( x 0 , y 0 ) and ( x 1 , y 1 ) are chosen, the ratio m =

y 1 − y 0 x 1 − x 0

will always be the same, which means

that the slope of a line is well-defined. That would not be true for any curve other than a straight line. The slope of a line measures how “slanted” the line is. For example, a slope of 0 means that the line is horizontal; a slope of 1 means that the line makes an angle of 45◦^ with the positive x -axis; and a slope of −1 means that the line makes an angle of 45◦^ with the negative x -axis.

Equation of a Line

Any line in the plane, whether vertical or not, can be given by an equation in x and y. There is a general form of the equation of a line that includes all lines, and then there are special forms for vertical lines and non-vertical lines.

Equation of a Line

1. The general form for the equation of a line is ax + cy = d.
2. The equation of a vertical line has the form x = p.
3. The equation of a non-vertical line has the form y = mx + b , where m is the slope of the line and

1.3. LINEAR FUNCTIONS 19

The equation we want to find has the form y = mx + b. Using the value of m that we found above, we see that this equation is y = 53 x + b. We now substitute ( x, y ) = (9 , 11) into that equation, which yields 11 = 53 · 9 + b. Solving for b we obtain b = −4. Hence the desired equation is y = 53 x − 4.

EXERCISES

1–4 Find the equation of each line.

1. The line containing the point (1 , 2) that has slope 3
2. The line containing the point (5 , 0) that has slope − 2
3. The line containing the point (2 , 1) that is parallel to the line y = 5 x − 3
4. The line containing the point (3 , 2) that is perpen- dicular to the line y = 2 x + 1

5–8 Find the equation of the line containing each pair of points.

5. (1 , 2) and (3 , 8) 6. (2 , 1) and (− 5 , 6)
6. (3 , 4) and (− 2 , 4) 8. (4 , 2) and (4 , −1)

9–12 For each pair of lines, state whether they are parallel, perpendicular or neither.

9. y = 2 x + 5 and 6 x − 3 y = 4
10. y = 3 x + 4 and x + 3 y − 1 = 0
11. x − 4 y + 2 = 0 and 2 x − 6 y = 5
12. y = 7 and x = 2

20 CONTENTS

1.4 (^) Polynomials

Linear functions are the simplest type of broadly useful functions, though of course not everything in the world is linear. The next simplest type of function is polynomial functions. Of course, all linear functions are polynomial functions, though not vice-versa.

Polynomial Functions

Some basic terminology about polynomial functions is the following.

Polynomial Functions

1. A polynomial function is a function f ( x ) that can be defined by a formula of the form f ( x ) = anxn^ + an − 1 xn −^1 + · · · + a 2 x^2 + a 1 x + a 0 , where an, an − 1 ,... , a 2 , a 1 , a 0 are real numbers, and where an , 0.
2. The coefficients of the polynomial function f ( x ) = anxn^ + an − 1 xn −^1 + · · · + a 2 x^2 + a 1 x + a 0 are the numbers an, an − 1 ,... , a 2 , a 1 , a 0.
3. The leading coefficient of the polynomial function f ( x ) = anxn^ + an − 1 xn −^1 + · · · + a 2 x^2 + a 1 x + a 0 is the number an , which is never zero.
4. The constant term of the polynomial function f ( x ) = anxn^ + an − 1 xn −^1 + · · · + a 2 x^2 + a 1 x + a 0 is the number a 0.
5. The degree of the polynomial function f ( x ) = anxn^ + an − 1 xn −^1 + · · · + a 2 x^2 + a 1 x + a 0 is the number n.

The domain of any polynomial function is the set of all real numbers. The range of every odd-degree polynomial function is the set of all real numbers (this fact is evident from the graphs of polynomials, though a proof is subtle). The range of every even-degree polynomial function is not the whole set of real number. More precisely, if the leading coefficient of an even-degree polynomial is positive, then the polynomial has a minimum value, and the the range consists of all real numbers greater than or equal to this minimum value; if the leading coefficient of an even-degree polynomial is negative, then the polynomial has a maximum value, and the the range consists of all real numbers less than or equal to this maximum value.

Graphs of Polynomial Functions

In general, the graph of a polynomial function of degree n is a curve that intersects the x -axis at most n times, and has at most n − 1 “bumps.” The graphs of polynomial functions do not have asymptotes (unless they are constant polynomials).

Graphs You Should Know

It is not possible to know the graph of every polynomial, but there are two that are used so often that they are worth knowing. Graphs you should know: y = x^2 and y = x^3.