Chapter 2-algebra

Question 1

What are the different categories of the set of real numbers?

Answer 1

• Natural numbers
• Whole numbers
• Integers
• Rational numbers
• Irrational numbers

These categories represent various types of numbers within the real number system.

Question 2

What is the set of real numbers (R) composed of?

Answer 2

• Natural numbers
• Whole numbers
• Integers
• Rational numbers
• Irrational numbers

Each category has its own definition and examples.

Question 3

Define Natural Numbers.

Answer 3

Contain all counting numbers which start from 1

Examples include: 1, 2, 3, 4, … ∞.

Question 4

Define Whole Numbers.

Answer 4

Collection of zero and natural numbers

Examples include: 0, 1, 2, 3, 4, … ∞.

Question 5

Define Integers.

Answer 5

The set of whole numbers and negatives of all natural numbers

Examples include: -∞, …, -4, -3, -2, -1, 0, 1, 2, 3, 4, … ∞.

Question 6

Define Rational Numbers.

Answer 6

A- Numbers that can be written in the form of P/q, where q ≠ 0
B- terminated or repeated decimals

Examples include: 0.3, 1.16, 0.23, 3.14, -5.8.

Question 7

Define Irrational Numbers.

Answer 7

All numbers which are not rational and cannot be written in the form of P/q

Examples include: √2, π, √25.

Question 8

What is the symbol for the Set of Real Numbers?

Answer 8

R

The set of real numbers includes all rational and irrational numbers.

Question 9

List the properties of real numbers that are true for any real numbers a, b, c, d, x, y, and z.

Answer 9

1. Distributive property.
2. Commutative property of addition and multiplication.
3. Associative property of addition and multiplication.
4. Additive inverse property.
5. Multiplicative inverse property.
6. Substitution property.
7. Addition and subtraction property of equality.
8. Multiplication and division property of equality.
9. Division property of equality.
10. transitive property of equality.

Question 10

What is a repeating decimal?

Answer 10

A decimal number in which one or more digits repeat forever in a pattern after the decimal point

Examples include 0.3 = 0.333… and 0.27 = 0.2727…

Question 11

Convert the repeating decimal 0.3 into a fraction.

Answer 11

0.3 = 3/9 = 1/3

The repeated decimal is 3, which is one digit.

Question 12

Convert the repeating decimal 0.27 into a fraction.

Answer 12

0.27 = 27/99 = 3/11

The repeated decimal is 27, which is two digits.

Question 13

What is interval notation?
And what are the types of intervals

Answer 13

A way of writing subsets of the real number line
Types :

Closed interval , open interval
Have open , half closed intervals
Intervals include infinity

It is written from left (least) to right (greatest).

Question 14

In interval notation, how is the inequality < represented?

Answer 14

Open interval ( ) and a open circle ( o )

It is represented by a circle (o) on the number line.

Question 15

In interval notation, how is the inequality ≤ represented?

Answer 15

Closed interval [ ] and a shaded circle (•)

It is represented by a dot (•) on the number line.

Question 16

State the description of an open interval?

Answer 16

Does not include its endpoints
Uses parenthesis

Question 17

State the description of an closed interval?

Answer 17

Includes its endpoints
Uses closed brackets []

Question 18

State the description of an ** half open and half closed intervals, **?

Answer 18

Includes only one endpoint
Uses combination of brackets and parentheses 

Question 19

State the description of ** intervals that include infinity **?

Answer 19

End points are negative infinity or positive infinity
Uses combination of brackets and parentheses as needed
→ (arrow) on the number line.

Question 20

What is the interval notation for the inequality -5 ≤ x ≤ 1?

Answer 20

[-5, 1]

This indicates that x includes both endpoints -5 and 1.

Question 21

What is the interval notation for the inequality -5 < x < 1?

Answer 21

(-5, 1)

This indicates that x does not include the endpoints -5 and 1.

Question 22

What is the interval notation for the inequality -5 ≤ x < 1?

Answer 22

[-5, 1)

This indicates that x includes -5 but not 1.

Question 23

What is the interval notation for the inequality x ≥ 2?

Answer 23

[2, ∞)

This indicates that x includes 2 and goes to positive infinity.

Question 24

What is the interval notation for the inequality x > 2?

Answer 24

(2, ∞)

This indicates that x does not include 2 and goes to positive infinity.

Question 25

What is the interval notation for the inequality **x ≤ 2**?

Answer 25

(-∞, 2] ## Footnote This indicates that x includes 2 and goes to negative infinity.

Question 26

What is the interval notation for the inequality **x < 2**?

Answer 26

(-∞, 2) ## Footnote This indicates that x does not include 2 and goes to negative infinity.

Question 27

What is the **Set Builder Notation** used for?

Answer 27

A mathematical shorthand for accurately stating a specific group of numbers ## Footnote It allows for concise representation of sets based on defined properties.

Question 28

Translate the following Set Builder Notation: {x € Z| - 4 ≤ x < 5}.

Answer 28

{-4, -3, -2, -1, 0, 1, 2, 3, 4} ## Footnote This notation describes the set of integers x that are greater than or equal to -4 and less than 5.

Question 29

In the Set Builder Notation, what does the symbol **€** represent?

Answer 29

Belongs to ## Footnote It indicates that the element is a member of the specified set.

Question 30

What does the vertical bar **|** signify in Set Builder Notation?

Answer 30

Such that ## Footnote It separates the variable from the condition that defines the set.

Question 31

Fill in the blank: The Set Builder Notation can be used to express sets of numbers based on specific _______.

Answer 31

properties ## Footnote This allows for flexibility in defining sets beyond simple enumeration.

Question 32

What is the **Set Builder Notation** used for?

Answer 32

A mathematical shorthand for accurately stating a specific group of numbers ## Footnote It allows for concise representation of sets based on defined properties.

Question 33

Translate the following Set Builder Notation: {x € Z| - 4 ≤ x < 5}.

Answer 33

{-4, -3, -2, -1, 0, 1, 2, 3, 4} ## Footnote This notation describes the set of integers x that are greater than or equal to -4 and less than 5.

Question 34

In the Set Builder Notation, what does the symbol **€** represent?

Answer 34

Belongs to ## Footnote It indicates that the element is a member of the specified set.

Question 35

What does the vertical bar **|** signify in Set Builder Notation?

Answer 35

Such that ## Footnote It separates the variable from the condition that defines the set.

Question 36

Fill in the blank: The Set Builder Notation can be used to express sets of numbers based on specific _______.

Answer 36

properties ## Footnote This allows for flexibility in defining sets beyond simple enumeration.

Question 37

What is a **function**?

Answer 37

A binary relation between two sets that associates each element of the domain to exactly one element of the range ## Footnote The first set is referred to as the domain, and the second set is the range.

Question 38

What does the **vertical line test** determine?

Answer 38

If a graph represents a function ## Footnote If any vertical line intersects the graph at more than one point, it does not represent a function.

Question 39

What is a **one-to-one function**?

Answer 39

A function where no two elements in the domain correspond to the same element in the range ## Footnote Each x in the domain has exactly one image y in the range.

Question 40

What does the **horizontal line test** determine?

Answer 40

If a graph represents a one-to-one function ## Footnote If any horizontal line intersects the graph at more than one point, it does not represent a one-to-one function.

Question 41

In the equation **y = 3x² + 4x**, what is the dependent variable?

Answer 41

y ## Footnote The dependent variable is the one whose value depends on the independent variable.

Question 42

In the equation ** y = 3x² + 4x**, what is the independent variable?

Answer 42

x ## Footnote The independent variable is the one that can be changed freely without depending on other variables.

Question 43

If **f(x) = 3x² + 4x**, what is the output when **x = 2**?

Answer 43

20 ## Footnote Calculated as f(2) = 3(2)² + 4(2) = 12 + 8.

Question 44

If **f(x) = 3x² + 4x**, what is the output when **x = -a**?

Answer 44

3a² - 4a.

Question 45

In the piecewise function , find f(2)

Answer 45

12

Question 46

In the piecewise function , find f(0)

Answer 46

Undefined

Question 47

And the peace wise function find f(3)

Answer 47

8

Question 48

What are the **five common operations** that can be performed on functions?

Answer 48

* Adding * Subtracting * Multiplying * Dividing * Composition of functions ## Footnote The composition of functions involves a function inside of another function.

Question 49

In the operation **(f + g)(x)**, what does it represent?

Answer 49

f(x) + g(x) ## Footnote Example: If f(x) = 2x + 4 and g(x) = x + 1, then (f + g)(x) = 3x + 5.

Question 50

Fill in the blank: The operation **(f - g)(x)** is equal to _______.

Answer 50

f(x) - g(x) ## Footnote Example: If f(x) = 2x + 4 and g(x) = x + 1, then (f - g)(x) = x + 3.

Question 51

What does the operation **(f x g)(x)** represent?

Answer 51

f(x) * g(x) ## Footnote Example: If f(x) = 2x + 4 and g(x) = x + 1, then (f x g)(x) = 2x^2 + 6x + 4.

Question 52

What is the notation for the **composition of functions**?

Answer 52

f(g(x)) ## Footnote This indicates that function g is applied first, followed by function f.

Question 53

In the operation **(f / g)(x)**, what does it represent?

Answer 53

f(x)/g(x)

Question 54

What does the symbol ** ⌊x⌋** represent?

Answer 54

The integer number less than or equal to x also called a step function ## Footnote Remember an integer is the set of whole numbers and negative of all natural numbers

Question 55

What is the value of ⌊5 ⌋

Answer 55

5 ## Footnote Still an integer, therefore we keep it the same

Question 56

What is the value of ⌊-4 ⌋

Answer 56

-4 ## Footnote Still an integer, therefore we keep it the same 

Question 57

What is the value of ⌊-2.9⌋

Answer 57

-3 ## Footnote Here this is not an integer, but it is a decimal therefore we must change it to an integer number less than or equal to -2.9 by -2.9 is a decimal so -3

Question 58

What is the value of ⌊2.9⌋

Answer 58

2 ## Footnote This is also a decimal that is why we must change it to an integer number less than or equal to 2.9 but we can’t do equal to 2.9 because it is a decimal so we choose something that is less therefore 2

Question 59

What is the range of **f(x) = ⌊x⌋ + 3**? A ) R B ) Z C ) [3,+ infinity ) D ) ( -infinity , 3 ]

Answer 59

B

Question 60

What is the domain of **f(x) = ⌊x⌋ + 3**? A ) R B ) Z C ) [3,+ infinity ) D ) ( -infinity , 3 ]

Answer 60

R ## Footnote The domain of the step function is all real numbers.

Question 61

What is the domain of the step function?

Answer 61

R ## Footnote This indicates that the domain consists of all real numbers.

Question 62

What is the Range of the step function?

Answer 62

Z ( integers )