Chapter 1 Functions

Chapter 1 Functions

1.1 Real Numbers, Inequalities and Lines

1.1.1 Real Numbers

Natural Number

Integer (Whole Number) 0

Rational Number Negative Number

Real Number (R) Fractional Number

Irrational Number

【Note】

In this book the word “number” means real number, a number that can be represented by a point on the number line (also called the real line).

1.1.2 Inequality

The order of the real numbers is expressed by inequalities. For example, means “a is to the left of ” or, equivalently, “b is to the right of ”. Summary results are as follows.

a<b

b a

【Note】

1. The inequalities a<b and a>b are called strict inequalities.

x<b hold. The inequality a< <x b can be interpreted graphically as “x is between a and b”.

1.1.3 Sets and Intervals

Braces { }are read “the set of all” and a vertical bar | is read “such that”.

【Example 1】

1.{^{x x| >3}} means “the set of all x such that x is greater than 3 ”.

2.{^{x| 2}− < <^{x 5}} means “the set of all x such that x is between –2 and 5”.

Finite Intervals

Infinite Intervals

1.1.4 Lines and Slopes

The symbol Δ (read “delta”) means “the change in”. For example any two points

1 1

( ,x y and ) ( ,x y we define _{2 2})

2 1

x x x

Δ = − The change in x is the difference in the x-coordinates.

2 1

y y y

Δ = − The change in y is the difference in the y -coordinates.

Slope of Line Through

【Definition 1】

Slope intercept from of a line

Point slope form of a line

Vertical Line and Horizontal Line

【Example 2】

1.Please find the slope of the line through each pair of points, and graph the line.

a.(2,1), (3, 4) b.(2, 4), (3,1) c.( 1− ,3), (2, 3) d.(2, 1)− , (2, 3)

2. Please find an equation of the line with slope –2 and y-intercept 4, and graph it.

3. Please find an equation for the line through the points (4,1) and (7,-2).

4. Please find an equation for the vertical line through (3,2).

5. Please find an equation for the horizontal line through (3,2).

【sol】

General linear equation

【Example 3】

Please find the slope and y-intercept of the line 2x+3y=12

【sol】

1.2 Exponents

Not all variables are related linearly. In this section, we will discuss exponents, which will enable us to express many nonlinear relationships.

【Definition 2】Positive integer exponent

Numbers may be expressed with exponents. More generally, for any positive integer ,

n x means the product of ⁿ n x′s.

n

xn = ⋅x x x

The number being raised to the power is called the base and the power is the exponent.

Exponent or power x n

base

There are several properties of exponents for simplifying expressions.

【Theorem1】Properties of exponents

【Definition 3】Zero and negative exponents For any numberxother than zero, we define 1.x⁰ =1

2. ¹ 1

x x

− = , ² 1₂

x x

− = , _n 1

x n

x

− =

3.

x 1 y

y x

⎛ ⎞− =

⎜ ⎟⎝ ⎠ ,

n n

x y

y x

⎛ ⎞− =⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠

⎝ ⎠

【Note】

0 and 0 are undefined. 0 ⁻ⁿ

【Example 4】

Please simplifying exponents

1.

2 3 4

5 7

( ) x ? x x x

⎡ ⎤

⎣ ⎦ =

⋅ ⋅ 2.( 2)− ⁻³ =?

【sol】

【Definition 4】Fractional exponents

Fractional exponents are defined as follows:

1.

1 n n

x = x power1

nmeans the principal nth root, where n is a positive integer.

2.^{xmn =}( )^{n x m= n xm xm n} means the th power of the th root, or equivalently, the th root of the th power.

m n

n m

【Note】

1. x± ≠y x± y 2.(x±y)² =x²±2xy+y2

3.(x±y)³ =x³±3x y² +3xy²±y³ 4.x²−y² =(x−y x)( +y)

5.x³−y³ =(x−y x)( ²+xy+y²) 6.x³+y³=(x+y x)( ²−xy+y²)

7.xⁿ −yⁿ =(x−y x)( ⁿ⁻¹+xⁿ⁻²y+ +xyⁿ⁻²+y^{n 1−})

【Example 5】

Please evaluate the follows fractional exponents.

1.

27 2 3

8

⎛− ⎞

⎜ ⎟

⎝ ⎠

2.

9 3/ 2

4

⎛ ⎞−

⎜ ⎟⎝ ⎠

【sol】

Learning curves in airplane production

It is a truism that the more you practice a task, the faster you can do it. Successive repetitions generally take less time, following a “learning curve”. Learning curves are used in industrial production. For example, it took 150000 work-hours to build the first Boeing 707 airliner, while later planes took less time.

Time

Repetitions

0.322

Time to build 150

n thousand work hours plane number n

⎛ ⎞ −

= −

⎜ ⎟

⎝ ⎠

10

n= ⇒ 150 (10)× ^−0.322 ≈71.46 thousand work-hours 100

n= ⇒ 150 (100)× ^−0.322 ≈34.05 thousand work-hours

1.3 Functions

In the previous section we saw that the time required to build a Boeing 707 airliner will vary, depending on the number that have already been built.

Mathematical relationships such as this, in which one number depends on another, are called functions, and are central to the study of calculus. In this section we defined and give some applications of functions.

【Definition 5】

A function f is a rule that assigns to each numberxin a set a number ( )f x . The set of all allowable values ofxis called the domain(定義域), and the set of all values ( )f x forxin the domain is called range(值域).

We will be mostly concerned with functions that are defined by formulas for calculating ( )f x fromx. If the domain of such a function is not stated, then it is always taken to be the largest set of numbers for which the function is defined, called the natural domain of the function. To graph a function f , we plot all points ( , )x y such thatxis in the domain and y= f x( ). We call x the independent variable and y the dependent variable, since y dependents onx.

【Example 6】

1.For the function 1

( ) 1

f x = x

− , please answer the following questions.

(1) Please find (5)f .

(2) Please find the domain of the function.

(3) Please find the range of the function.

2.For the functionf x( )=2x²+4x− , please answer the following questions. 5 (1) Please find f( 3)− .

(2) Please find the domain of the function.

(3) Please find the range of the function.

【sol】

【Note】

For each x in the domain of a function there must be a single number y= f x( ), so the graph of a function cannot have two points ( , )x y with the same x-value but different y -values. This leads to the following graphical test for functions.

Vertical line test for functions

【Definition 6】Linear function

A linear function is a function that can be expressed in the form ( )

f x =mx b+

with constantsmandb. Its graph is a line with slope m and y -intercept b.

【Example 7】

An electronic company manufactures pocket(袖珍型) calculators at a cost of $9 each, and the company’s fixed costs(such as rent(租金)) amount to $400 per day.

Please find a functionC x( ) that gives the total cost of producing x pocket calculators in a day.

【sol】

【Definition 7】Quadratic function

【Definition 8】Vertex formula for a parabola(抛物線)

【Definition 9】Solutions for quadratic formula

【Note】

The quantity , whose square root appears in the quadratic formula, is called the discriminant(判別式).

2 4

b a

Δ = − c

0

If Δ =b²−4ac> , the equation ax²+bx+ = has two real roots. c 0 If Δ =b²−4ac= 0, the equation ax²+bx+ = has only one root. c 0 If Δ =b²−4ac< 0, the equation ax²+bx+ = has no real root.. c 0

【Example 8】

A company that installs automobile compact disc(CD) players finds that if it installs x CD players per day, then its costs will be C x( ) 120= x+4800 and its revenue will be R x( )= −2x²+400x (both in dollars).

1.Please find the company’s break-even points.

2.Please find the number of units that maximizes profit, and the maximum profit.

【sol】

1.4 Functions, Continued

In this section, we will define other useful types of functions and an important operation, the composition of two functions.

【Definition 10】 Polynomial function

A polynomial function (or simply a polynomial) is a function that can be written in the form

1 2

1 2 1

( ) _n ⁿ _n ⁿ ₀

f x =a x +a ₋x ⁻ + +a x +a x+ a

where is a nonnegative integer and are real numbers, called coefficients. The domain of a polynomial is , the set of all real numbers. The degree of a polynomial is the highest power of the variable.

n a a₀, ₁, ,a_n

【Definition 11】Rational function

The word ”ratio” means fraction or quotient, and the quotient of two polynomials is called a rational function. The domain of a rational function is the set of all numbers for which the denominator is not zero.

【Example 9】

What is the domain of 18 ( ) ( 2)( 4

f x = x x

+ − )？

【sol】

【Note】

Simplifying a rational function by canceling a common factor from the numerator and the denominator can change the domain of the function, so that the “simplified”

and “original” versions may not be equal (since they have different domains).

For example:

2 1

( ) 1

f x x x

= −

−

【Definition 12】Exponential functions

A function in which the independent variable appears in the exponent is called an exponential function.

【Note】

1.Exponential functions are often used to model population growth and decline(衰 退).

2.In mathematics the letter is used to represent a constant whose value is approximately 2.718. The exponential function

e

( ) ^x

f x = will be very important e beginning in Chapter 4. Another important function is the logarithmic function to the base e, written f x( )=lnx.

【Definition 13】Piecewise linear functions

The rule for calculating the values of a function may be given in several parts. If each part is linear, the function is called a piecewise linear function, and its graph consists of “pieces” of straight lines.

【Example 10】

Graph 5 2 2

( ) 3 2

x if x

f x x if x

− ≥

= ⎨⎧⎩ + <

【sol】

【Note】The absolute value function

An important piecewise linear function is the absolute value function. The absolute value function is f x( )= x defined as

, 0

( ) , 0

x if x

f x x if x

⎧ ≥

= ⎨⎩− <

Graph:

【Definition 14】Composite function

The composition of f withgevaluated atxis ( ( ))f g x .

【Note】

The domain of f g x is the set of all numbers( ( )) xin the domain of such that is in the domain of

g ( )

g x f .

【Example 11】

If 8

( ) 1

f x x x

= +

− and ( )g x = x, please find (1) ( ( ))f g x

(2)g f x( ( ))

【sol】

【Example 12】

A planning commission estimates that if a city’s population is p thousand people, its daily water usage will be thousand gallons. The commission further predicts that the population in t years will be

( ) 30 1.2

W p = p

( ) 60 2

p t = + thousand t people. Express the water usageWas a function of , the number of years from now, and find the water usage 10 years from now.

t

【sol】

Shifts of graphs

Function Shift ( )

y= f x + a Shifted up by a units ( )

y= f x − a Shifted down by a units

( )

y= f x+a Shifted left by a units

( )

y= f x a− Shifted right by a units

【Definition 15】Difference quotients

The quantity f x( h) f x( ) h

+ − (h≠0) will be very important in Chapter 2 when we

begin studying calculus. It is called the difference quotient, since it is a quotient whose numerator is a difference. It gives the slope (rise over run) between the points in the curvey= f x( )atxand atx+h.

【Example 13】

1. If f x( )=x² −4x+ , please find and simplify 1 f x( h) f x( ) h

+ − , (h≠0)

2. If 1 ( )

f x = , please find and simplify x f x( h) f x( ) h

+ − , (h≠0)

【sol】

【補充】Trigonometric Functions (請見 Chapter 8 )

1.5 Triangles, Angles and Radian measure

【Definition 16】

【Note】

Angles and Radian Measure

Trigonometry requires a precise definition of the word angle. An angle is formed when a line segment is rotated around one of its endpoints (the vertex), from an initial side to a terminal (末端) side. We use an arrow (箭頭) to indicate the rotation (旋轉).

【Definition 17】Radian(弳度、弧度) Measure

【Note】

【Example 14】

(2) (3)

2. Please find the degree measure of each angle.

(1)

1. Please find the radian measure of each angle.

(1) 45 150 −60

radians (2) 3 4

π 6

π radians (3)

2

− radians (4) 1 radians π

【sol】

【Note】

(1) (initial ray) 線(terminal ray)

。

一個角(angle)有三部分：一條起始線 、一條終點

及一個頂點(vertex) 起始線與正x軸重疊且頂點在原點上，則稱角在標準位

tandard position)上。

) 銳角(acute angle)、直角(right angle)、鈍角(obtuse angle)、平角(straight angle)。

) 起始線逆時鐘方向(counterclockwise)的角為正，順時鐘方向(clockwise)的角 為負。同一 起始線，有相同終點線的角為同界角(cotermial)。

(4) 角度可超過 ，亦可低於 。

(5) 半徑(radius)為1 的圓，圓心角 置(s

(2

(3

360 −360

θ 的弳度(radian)就是圓心角θ 所對應扇形 (sector)的弧長。

因為圓周長( circumference )=2π(1)=2π，故有

(a) radians（弳；弧度）或 radians

(b)

π 2

360 = 180 =π

180 360

1 2π π

=

= radians ⇒

180

× π

= x

x radians

1 radian π 2

= 360 radian

π 2

×360

= y ⇒ y

(6) 三角形(triangle)的相關法則摘要：

1. 三角形的三個內角和為 。

2. 直角三角形的兩個銳角和為 。

3. 畢氏定理(Pythagorean Theorem)：直角三角形的兩股(leg)平方和等於斜 邊 (hypotenuse)的平方。

4. 相似三角形(similar triangles)：兩三角形的對應角相等，對應邊成比例。

5. 三角形的面積等於底乘以高的一半，亦即

180 90

bh

A 2

=1 。 6. 等邊三角形(equilateral triangle)的每一個內角皆為 。

7. 等腰直角三角形(isosceles right triangle)的每一個銳角皆為 。 8. 三角形的頂垂線(altitude)將底邊分成兩段。

60

45

【Definition 18】

【Note】

【Definition 19】

【E

The cities of Washington, D.C., and Moscow determine a central angle of 1.22 radians (at the center of the earth). Please find the distance between these cities measured along the surface of the earth. Assume that the earth is a sphere(球體) of radius 4000 miles.

【sol】

1.6 Sine and Cosine Functions

In this section, we define the sine and cosine functions, and use them to model uantities that fluctuate(波動) periodically(週期性).

【Definition 20】

【Example 16】

1. Please evaluate xample 15】

q

sinθ and cosθ for following values of θ : (1) 0 (2)

2

π (3) π (4) 3 2 π

、 2

sin π

and 2 cos π sinπ

、 cosπ 2.Please find

3.Please find sin π 4

⎛− ⎞

⎜ ⎟ and

⎝ ⎠ cos π

4

⎛− ⎞

⎜ ⎟

【

Sine and Cosine of an Acute Angle

An acute angle is an angle between 0 and

⎝ ⎠

sol】

π 2 radians. For an acute angleθ, sinθ and cosθ can be expressed as ratios of sides of a right triangle (called hyp,opp, and adj for the hypotenuse(斜邊), opposite side(對邊), and adjacent side (鄰 邊)

【Example 17】

How long a ladder (梯子) is needed to reach a roof (屋頂) 20 feet above the ground if the ladder makes a angle with the ground?

).

60

【sol】

Graphs of the Sine and Cosine Functions

Modified Sine and Cosine Curves－Changing Amplitude(振幅) and

Period(週期)

The amplitude of a sine or cosine curve is half the distance between its highest and lowest values (sometimes called its half-height). Therefore, and have amplitude 1. By multiplying by constants, we may create similar “wavy”

functions with any amplitude and period.

sin t cost

【Note】

的最大值為 1，最小值為

1. f(x)=sinx − ，振幅(amplitude)定義為最大值與最小1

值之差的一半，故 f(x)=sinx的振幅為 1。函數的週期(peroid)為x軸上相鄰兩 循環之間的距離，故 f(x)=sinx的週期為2π ，亦即sin(x±2nπ)=sinx。 2.y=asinbx與y=acosbx的圖形

皆在 與 之間擺動，故振幅皆為 ；

時， ；

−a a | a|

，故週期為

|

| 0 2 2

b b

π π _{− =}

=0

bx x=0 bx=2π時，

x= 2bπ 。

【Example 18】

Sketch the graph of f t( )=3sin 2t

【sol】

Seasonal Sales

The sine and cosine functions with modified amplitude and period are often used to model quantities that fluctuate in a regular pattern, such as seasonal sales.

【Example 6】

A travel agency specializing in Caribbean vacations models its daily revenue (in dollars) on day of the year by n

( ) 600 500 cos 2 365

R n _⎛ πn_⎞

= + ⎜ ⎟

⎝ ⎠.

60 n= Please find the company’s revenue on March 1 (day )

【sol】

【Theorem 2】Trigonometric Identities

In this section, we define the tangent, cotangent, secant, and cosecant functions, differentiate and integrate them, and discuss some of their applications.

【Definition 21】

【Example 19】

1. Please find (1) tanπ (2) cotπ (3) secπ (4) cscπ 2. Please find (1) tan

3

π (2) cot 3

π (3) sec 3

π (4) csc 3 π

【sol】

【Definition 22】

【Note】Graphs of Tangent, Cotangent, Secant and Cosecant Functions

【Note】

1. .

csc . .

sin .

opp hyp hyp

opp =

= θ

θ

. sec .

. cos .

adj hyp hyp

adj =

= θ

θ

. cot .

. tan .

opp adj adj

opp =

= θ

θ

y r r

y =

= θ

θ csc

sin 2.

x r r

x =

= θ

θ sec

cos

y x x

y =

= θ

θ cot

tan

θ θ

θ θ θ θ

θ ¹θ sin =

3. csc cot

1 cos

tan sin sec

cos = 1 = =

θ

θ θ

θ θ

θ ¹

n cot cos

c sec 1

csc = 1 = = =

4.Trigonometric identities(三角恆等式)

(1) Pythagorean Identities（畢氏恆等式）

(2) Reduction Formulas（簡化公式）

θ

θ os si tan

sin

1 cos sin²θ + ²θ =

θ

θ ²

2 1 sec

tan + = θ

θ ²

2 1 csc

cot + =

θ θ) sin sin(− =− cos(−θ)=cosθ tan(−θ)=−tanθ sinθ =−sin(θ −π) cosθ =−cos(θ −π) tanθ =tan(θ −π)

(3) Sum or Difference of Two Angles sin(θ ±φ)=sinθcosφ±cosθsinφ θ φ sθcosφ ∓sinθsinφ

co ) cos( ± =

φ θ

φ φ θ

θ 1 tan tan tan ) tan

tan( ∓

= ±

±

(4)Double Angle Formula（倍角公式）

sin(2θ)=2sinθcosθ

(5)Half Angle Formula（半角公式）

θ θ

θ θ

θ) cos² sin² 2cos² 1 1 2sin² 2

cos( = − = − = −

2 2 cos sin²θ =1⁻ θ

2 cos cos²θ =1⁺ θ

5.角度可以用度數或弳度表示，微積分中偏愛使用弳度，因此除非特別聲明，角

度皆以弳度表示。例如 ，一個是弳度 3，一個是度數 3。

6. 與

3 sin 3 sin ≠

的圖形，週期皆為

|

| b

= π bx

a

y= tan y=acotbx 。

與 的圖形，週期皆為

|

| 2

b

= π bx

a

y= csc y=asecbx 。