MATH 1010 Note 2-Functions PDF

Preview

Summary

Dr. Jeff Chak-Fu WONGLecture NoteFunctions 2 :Chinese University of Hong KongDepartment of Mathematicsjwongmath.uhk.eduProduced by Jeff Chak-Fu WONGUniversity MathematicsMATH 10101➁➀➂➃ Elementary FunctionsSetsComposite Functions and Inverse FunctionsFunctions TABLE OF CONTENTTABLE OFCONTENT 2DEFINI...

Description

Lecture Note 2: Functions Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong

MATH1010 University Mathematics Produced by Jeff Chak-Fu WONG

1

T ABLE

OF

C ONTENT

Sets Functions Composite Functions and Inverse Functions Elementary Functions

TABLE OF C ONTENT

2

S ETS • Definition of Set • Operations upon Sets • The Set of Real Numbers

S ET S

3

D EFINITION

D EFINITION OF S ET

OF

S ET

4

Definition 1

• A set is a collection of objects that is itself considered as an entity. • The object may have any character so long as we know which objects are in a given set and which are not.

Example 1 A = {1, 2, 3, 4, 5} means A is the set consisting of numbers 1, 2, 3, 4 and 5.  D EFINITION OF S ET

5

Remark 1

• We usually use a capital letter A, B , S or T to denote a set. • A set can be described by using a pair of curly braces { symbols.

} with words or

• We can also write the set A as {x|x is a natural number less than 6}. In this notation, 1. the symbol x indicates a variable and stands for a general element of the set, 2. the vertical bar | stands for the words “such that”, and 3. the properties which determine the membership in the set are written to the right of the vertical bar.

D EFINITION OF S ET

6

• If A is the set and p is an object in A, we write: p ∈ A, and say p is an element of A or p is in A. • The notation p∈A / indicates that p is not an element of A.

Example 2 A = {x| x is an odd number, x < 10} and B = {2, 3, 5, 7, 15, 21}. Observe that – 9 ∈ A but 9 ∈ / B, – 21 ∈ B but 21 ∈ / A, – 3 ∈ A and 3 ∈ B, and – 6∈ / A and 6 ∈ / B. 

D EFINITION OF S ET

7

•

– If a set has a finite element of elements, it is called a finite set, otherwise it is called an infinite set. – For a special case in which a set contains no element, we say the set is empty and use the symbol ∅ to denote the empty set.

Example 3 Let S = {x| x2 = 9, x is even}.

Then S = ∅; that is, S is the empty set. 

D EFINITION OF S ET

8

•

– If every element of a set A is also an element of a set B , then A is a subset of B, and is denoted by A⊆B. That is, if x ∈ A, then x in B. In particular, every set is a subset of itself. If it is not true that A is a subset of B, we write A * B or A ( B. Thus, A * B or A ( B if there is an element of A that is not in B.

Figure 1: A is a subset of B .

D EFINITION OF S ET

9

– If A and B are sets, then we say that A equals B, written A = B, whenever, for any x, x ∈ A if and only if x ∈ B. An alternative definition is that A = B if and only if A⊆B and B⊆A. – If A⊆B and A 6= B, we write A ⊂ B and say that A is a proper subset of B .

D EFINITION OF S ET

10

Figure 2: Sunny Side Up Egg

D EFINITION OF S ET

11

Example 4 We use the following special symbols: – N = the set of positive integers: 1, 2, 3, · · ·

– Z = the set of integers: · · · , −2, −1, 0, 1, 2, · · · – R = the set of real numbers.

Thus we have N ⊂ Z ⊂ R. The hierarchy of numbers is summarized in Figure 3.

D EFINITION OF S ET

12

Figure 3: The hierarchy of numbers. 

D EFINITION OF S ET

13

O PERATIONS

O PERAT IONS UPON S ETS

UPON

S ETS

14

If A and B are sets, the union, A ∪ B , and the intersection, A ∩ B , are defined by Union Intersection

O PERAT IONS UPON S ETS

A ∪ B = {x| x ∈ A or x ∈ B},

A ∩ B = {x| x ∈ A and x ∈ B}.

15

Thus, • the union (cf. Figure 4) consists of the elements which are either in A, in B or in both A and B, and

Figure 4: A ∪ B is shaded. The above diagram, called Venn diagrams, illustrated the above-mentioned set operation. Here sets are represented by simple plane area, and the universal set U , by the area in the entire rectangle.

O PERAT IONS UPON S ETS

16

• the intersection (cf. Figure 5) consists of the elements that are both A and B; in other words, A ∩ B is the common part of A and B .

Figure 5: A ∩ B is shaded. The above diagram, called Venn diagrams, illustrated the above-mentioned set operation. Here sets are represented by simple plane area, and the universal set U , by the area in the entire rectangle.

O PERAT IONS UPON S ETS

17

It may happen that two sets A and B have no element in common. In such case, we say that they are disjoint, or their intersection is empty; that is, A ∩ B = ∅.

O PERAT IONS UPON S ETS

18

• Let U be a set and A a subset of U . • By the complement of A ∈ U such that x does not lie in A. ′

• We usually employ the notation Ac or A (cf. Figure 6), to denote this set. That is, ′ A = U \ A. • By definition, it is obvious that the following two statements are true: ′

(1)

A ∪ A = U, ′

A ∩ A = ∅.

(2)

′

Figure 6: A is shaded.

O PERAT IONS UPON S ETS

19

The difference of A and B or the relative complement of B with respect to A, denoted by A \ B (cf. Figure 7), is the set of elements which belong to A but not to B: A \ B = {x| x ∈ A, x ∈ / B}.

(3)

Figure 7: A \ B is shaded.

O PERAT IONS UPON S ETS

20

Example 5 Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, where U = {1, 2, 3, 4, · · · }. Then • A ∪ B = {1, 2, 3, 4, 5, 6}. • A ∩ B = {3, 4}. • A \ B = {1, 2}. ′

• A = {5, 6, 7, · · · }. 

O PERAT IONS UPON S ETS

21

T HE S ET

T HE S ET OF R EAL N UMBERS

OF

R EAL N UMBERS

22

T HE S ET OF R EAL N UMBERS

23

First, we identify the set of real numbers R with the axis, and the call the axis the number line, as shown in Figure 8.

Figure 8: The number line. In the following, we shall define major important subsets of R, called intervals.

T HE S ET OF R EAL N UMBERS

24

• The set of all numbers x satisfying the inequality a < x < b is called an open interval and is denoted by (a, b); that is, (a, b) = {x| a < x < b}.

(4)

• The closed interval from a to b is the open interval (a, b) together with the two end-points a and b, and is denoted by [a, b]; that is, [a, b] = {x| a ≤ x ≤ b}.

(5)

• The half-open interval (a, b] and [a, b) are defined by (a, b] = {x| a < x ≤ b}

(6)

[a, b) = {x| a ≤ x < b}.

(7)

and

T HE S ET OF R EAL N UMBERS

25

T HE S ET OF R EAL N UMBERS

26

• We use the symbols “ + ∞” and “ − ∞” to denote “positive infinity” and “negative infinity”, respectively.

They are only notations and not to be interpreted as representing any real number. Infinite intervals are defined as follows: (a, +∞) = {x| x > a},

(8)

(−∞, b) = {x| x < b},

(9)

[a, +∞) = {x| x ≥ a},

(10)

(−∞, b] = {x| x ≤ b},

(11)

(−∞, +∞) = R.

(12)

T HE S ET OF R EAL N UMBERS

27

Figure 9: A solid dot • means “including” and an open dot ◦ means “not including”.

T HE S ET OF R EAL N UMBERS

28

F UNCTIONS • Definition • Some Properties of Functions

FUNCTIONS

29

D EFINITION

D EFINITION OF FUNCTIONS

OF

F UNCTIONS

30

Implicit Form of a Function In general, when a function f is defined by an equation in x and y, we say that the function f is given implicitly. If is possible to solve the equation for y in terms of x, then we write y = f (x) and say that the function is given explicitly. For example, Implicit Form

ExplicitForm

3x + y = 5

y = f (x) = −3x + 5

32 − y = 0 xy = 4

D EFINITION OF FUNCTIONS

y = f (x) = x2 − 6 4 y = f (x) = x

31

Definition 2 A function f from a subset D of R to a subset E of R is a correspondence that assigns each element x of D to exactly one element y of E .

f :

D

−→

E

⊆

⊆

R

R

x

7−→

y = f (x)

Remark 2 • The element y of E is called the value of f at x (or the image of x under f ) and is denoted by f (x) if y corresponds to “an” element x of D . • x is sometimes also called an independent variable and y is called a dependent variable. D EFINITION OF FUNCTIONS

32

f :

D

−→

E

⊆

⊆

R

R

x

7−→

y = f (x)

• The set D in this definition is the domain of the function f , and the set E is the co-domain of f . • The range of f is the subset of E consisting of all possible function values f (x) for x in D .

D EFINITION OF FUNCTIONS

33

Domain

Rule

Range

D EFINITION OF FUNCTIONS

34

Domain x Rule f f(x ) Range

D EFINITION OF FUNCTIONS

35

Domain, Codomain and Range There are special name for what can go into, and what can come out of a function: • What can go into a function is called the Domain • What may possibly come out of a function is called the Co-domain • What actually comes out of a function is called the Range.

D EFINITION OF FUNCTIONS

36

Example 6 Consider the following simple example:

We note that • the set “A” is the Domain, • the set “B” is the Co-domain, • and the set of elements that get pointed to in B (the actual values produced by the function) are the Range, also called the Image. D EFINITION OF FUNCTIONS

37

We conclude that • Domain: {1, 2, 3, 4} • Co-domain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } • Range: {3, 5, 7, 9} 

D EFINITION OF FUNCTIONS

38

Example 7 Define a function f (x) = 2x with a domain and co-domain of integers (because we say so). That is, f :

D Z x

E :=

:=

−→

Z 7−→

y = f (x) = 2x

But by thinking about it we can see that the range (actual output values) would be just the even integers. So the co-domain is integers (we defined it that way), but the range is even integers. That is, { 2k| k ∈ Z} (set of all even integers) ⊂ E = Z 

D EFINITION OF FUNCTIONS

39

Functions can usually be represented in at least three different ways: 1. by tables, 2. by graphs, or 3. by formulas (or formulae).

D EFINITION OF FUNCTIONS

40

Importance of the domain of a function If x is in the domain of a function f, we shall say that f is defined at x, or f (x) exists. If x is not in the domain of f , we say that f is not defined at x, or f (x) does not exist. The value of the function f (x) at x = a (i.e. f (a)) is finite which means that the value of the function f (x) at x = a should not be anyone of the following forms: 0 0 ∞ , 0 , 0 × ∞, , ∞ − ∞, ∞0 , 1∞ , 0 ∞ imaginary value,

D EFINITION OF FUNCTIONS

a real number . 0

41

Example 8 If f (x) = (Do you see why?)

x , then f (0) exists. But f (1) and f (−1) do not exist. x2 − 1

 10

8

6

4

y

2

0

−2

−4

−6

−8

−10 −10

−8

−6

−4

−2

0

2

4

6

8

10

x

D EFINITION OF FUNCTIONS

42

−1.01

x f (x) =

−1.0001

−1.00001

→

−1

(x − 1)(x + 1)

x f (x) =

−1.001

x

−1

←

−0.99999

−0.9999

−0.999

−0.99

x (x − 1)(x + 1)

D EFINITION OF FUNCTIONS

43

0.99

x f (x) =

0.9999

0.99999

→

1

(x − 1)(x + 1) 1

x f (x) =

0.999

x

←

1.00001

1.0001

1.001

1.01

x (x − 1)(x + 1)

D EFINITION OF FUNCTIONS

44

Example 9 - If a function f is stated by a formula or rule for f (x) , i.e., f (x) =

1 , x

the domain is then assumed to be the set of all real numbers such that f (x) is real. 10

8

6

4

y

2

0

−2

−4

−6

−8

−10 −10

−8

−6

−4

−2

0

2

4

6

8

10

x

- Thus, for f (x) =

D EFINITION OF FUNCTIONS

1 , the domain of f is (−∞, 0) ∪ (0, +∞). x 45



D EFINITION OF FUNCTIONS

46

−0.01

x f (x) =

−0.001

−0.0001

−0.00001

→

0

←

0.00001

0.0001

0.001

1 x

D EFINITION OF FUNCTIONS

47

0.01

• One-to-one: – A function is a one-to-one function if f (x) 6= f (y) whenever x 6= y . – A special kind of one-to-one function is the identity function: ∗ a function I with the property that I(x) = x for all x in its domain.

D EFINITION OF FUNCTIONS

48

D EFINITION OF FUNCTIONS

49

• Graph of a function f :

– Let f be a function with domain D . – We associate each x in D with its exact function value f (x), and then get an ordered pair of numbers (x, f (x)). – In the two dimensional coordinate plane, (x, f (x)) denotes a unique point in the plane. – The graph of the function f is the set of all the points (x, f (x)), where x is in D; that is, the set {(x, f (x))| x ∈ D }.

D EFINITION OF FUNCTIONS

50

Example 10 Given a function f (x) = and sketch its graph.

√

1 − x2 , find the domain and the range of f ,

Solution: • Domain of f

– The domain of f is the set of all real numbers such that f (x) is real. – As f (x) exists if and only if the radicand 1 − x2 is non-negative; that is, 1 − x2 ≥ 0, or equivalently, −1 ≤ x ≤ 1, the domain is [−1, 1].

D EFINITION OF FUNCTIONS

51

• Range of f

– For any real number x in [−1, 1], the function value f (x) is given by

√

1 − x2 .

– As 0 ≤ 1 − x2 ≤ 1, whenever x is in [−1, 1],

(13)

it follows that 0 ≤ 1 − x2 ≤ 1.

(14)

– Hence, the range is [0, 1].

D EFINITION OF FUNCTIONS

52

• The graph of f is the upper portion of the unit circle, as illustrated in Figure 10. 

Figure 10: f (x) =

D EFINITION OF FUNCTIONS

√

1 − x2 .

53

Example 11 Find the domain of the given function f (x) =

x4 x2 + x − 6

Solution:

D EFINITION OF FUNCTIONS

54

Solution: The function f (x) =

x4 x2 + x − 6

is defined for all x except when 0 = x2 + 6 − x = (x + 3)(x − 2) ⇐⇒ x = −3 or 2, so the domain is {x ∈ R|x = 6 −3, 2}. 50

40

30

20

y

10

0

−10

−20

−30

−40

−50 −4

−3

−2

−1

0

1

2

3

4

x

Figure 11: Graph of f (x) = D EFINITION OF FUNCTIONS

x2 , where x ∈ [−4, 4]. x2 + x − 6 55



D EFINITION OF FUNCTIONS

56

Functions that are described by more than one expression, as in the next example, are called piecewise-defined functions. Example 12 Sketch the graph of the function defined as follows:

f (x) =

   x + 1, x2 ,   1,

D EFINITION OF FUNCTIONS

if x < 0, if 0 ≤ x ≤ 2,

(15)

if x > 2.

57

Solution:

  x + 1, if x < 0,  f (x) = x2 , if 0 ≤ x ≤ 2,    1, if x > 2.

1. If x < 0, then f (x) = x + 1, and the graph of f is part of the line y = x + 1, as shown in Figure 12. The open circle indicates that the point (0, 1) is not on the graph. 2. If 0 ≤ x ≤ 2, then f (x) = x2 , and the graph of f is part of the parabola y = x2 . 3. If x > 2, the function values are a constant number 1, and the graph of f is horizontal half-line without the end-point (2, 1). 

D EFINITION OF FUNCTIONS

58

5

•

4

3

y

2

1

•

0

−1

−2

−3 −3

−2

−1

0

1

2

3

4

5

x

Figure 12: Graph of Example 12

D EFINITION OF FUNCTIONS

59

- In Example 12, we observe that the graph of a piecewise-defined function is usually made up of several disconnected pieces. - Another function with this property is the greatest integer function f defined by f (x) = [x], where [x] is the greatest integer function, for instance, • [4] = 4, • [4.8] = 4, • [π] = 3,

• [− 21 ] = −1.

D EFINITION OF FUNCTIONS

60

Example 13 Find all function values of f (x) = [x] for x in the interval [−2, 2], and sketch the graph of the greatest integer function. Solution: According to the definition of the greatest integer function, we list the x and its corresponding function value f (x), where x is in [−2, 2], in Table 1.

value of x

f (x) = [x]

−2 ≤ x < −1

−2

0≤x...