Mathematics Chapter 1: Real Numbers
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Questions and Answers

What is the interval notation for the set of all real numbers greater than 5?

(5, ∞)

Express the absolute value of a real number 'a' in a mathematical way.

|a|

Given the piecewise function defined as y = -x for x > 0 and y = x + 1 for x ≤ 0, what is the range of this function?

(-∞, 1] ∪ (0, ∞)

What is the interval notation for the set of all numbers between -3 and 2, excluding -3 and 2?

<p>(-3, 2)</p> Signup and view all the answers

For the function f(x) = |x + 2|, write the equivalent piecewise function.

<p>f(x) = x + 2 if x + 2 ≥ 0; f(x) = -(x + 2) if x + 2 &lt; 0</p> Signup and view all the answers

What is the domain of the piecewise function y = -x for x > 0 and x + 1 for x ≤ 0?

<p>All real numbers</p> Signup and view all the answers

Rewrite the function f(x) = |x - 1| as a piecewise function.

<p>f(x) = x - 1 if x - 1 ≥ 0; f(x) = -(x - 1) if x - 1 &lt; 0</p> Signup and view all the answers

Describe the geometric interpretation of absolute value in terms of distance.

<p>The absolute value represents the distance from a number to 0 on the number line.</p> Signup and view all the answers

Define real numbers and provide an example.

<p>Real numbers include all rational and irrational numbers, such as $5$ or $ rac{1}{2}$. An example of an irrational number is $ rac{ ext{sqrt}(2)}{3}$.</p> Signup and view all the answers

What are the properties of real numbers that are crucial for arithmetic operations?

<p>The key properties of real numbers include the commutative, associative, and distributive properties. These properties aid in simplifying calculations.</p> Signup and view all the answers

Explain the difference between open and closed intervals.

<p>An open interval, such as $(0,7)$, excludes the endpoints, while a closed interval, like $[0,7]$, includes both endpoints. This also affects how intervals are represented graphically.</p> Signup and view all the answers

What is the interval notation for the set of all integers greater than 0 and less than 7?

<p>The interval notation is $(0, 7)$, meaning it includes all real numbers between 0 and 7, excluding the endpoints. Alternatively, using set-builder notation, it can be written as $ ext{A} = { x | x ext{ is an integer and } 0 &lt; x &lt; 7 }$.</p> Signup and view all the answers

What does the union of two sets represent?

<p>The union of two sets, denoted as $S igcup T$, represents all elements that are in either set or both. It combines the contents without duplicates.</p> Signup and view all the answers

Identify and explain the significance of the empty set.

<p>The empty set, denoted by $ ext{∅}$, contains no elements and is significant as it serves as the identity element for the union operation. It plays a crucial role in set theory.</p> Signup and view all the answers

How do repeating decimals relate to rational numbers?

<p>Repeating decimals are representations of rational numbers, which can be expressed as a fraction of two integers. For example, $0.333... = rac{1}{3}$.</p> Signup and view all the answers

What is the significance of absolute value in mathematics?

<p>Absolute value indicates the distance of a number from zero on the number line, disregarding its sign. For instance, $| -7 | = 7$.</p> Signup and view all the answers

How can the absolute value function be represented as a piecewise function?

<p>An absolute value function can be represented as a piecewise function with two cases: one for values greater than or equal to zero and another for values less than zero.</p> Signup and view all the answers

Why is x = 3 considered a breakpoint in the piecewise representation of an absolute value function?

<p>x = 3 is the breakpoint because it is the value at which the function changes its definition, transitioning from one case to another.</p> Signup and view all the answers

For the absolute value function rewritten at x > 3, which expression should be used: (x - 3) or (3 - x)?

<p>(x - 3) should be used for x &gt; 3.</p> Signup and view all the answers

What happens to the piecewise representation if the breakpoint changes from x = 3 to x = 7?

<p>The piecewise representation will change to reflect the new breakpoint at x = 7, altering the constants in the function.</p> Signup and view all the answers

When given a function with a breakpoint, how should one test the accuracy of the piecewise function?

<p>One should test the function using various values around and at the breakpoint.</p> Signup and view all the answers

Express the absolute value function |x - 3| as a piecewise function.

<p>|x - 3| = { x - 3, for x ≥ 3; 3 - x, for x &lt; 3 }</p> Signup and view all the answers

What is the general approach for rewriting any absolute value function into a piecewise function?

<p>Identify the breakpoint, then define two cases based on whether the input is greater than or less than this breakpoint.</p> Signup and view all the answers

Flashcards

Real Numbers

The set of all numbers, including rational and irrational numbers.

Rational Numbers

Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero.

Irrational Numbers

Numbers that cannot be expressed as a simple fraction of integers.

Set Builder Notation

Describes a set by stating the properties that its members must satisfy.

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Open Interval

An interval that does not include its endpoints.

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Closed Interval

An interval that includes its endpoints.

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Union of Sets

The set of all elements in either of two sets or in both.

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Intersection of Sets

The set of all elements that are contained in both of two sets.

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Interval Notation

A way to represent a set of numbers using parentheses and brackets to indicate whether the endpoints are included or excluded.

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Absolute Value

The distance of a number from zero, always non-negative.

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Piecewise Function

A function defined by multiple sub-functions, each applied to a specific interval of the domain.

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Graphing a Piecewise Function

To graph a piecewise function, you graph each sub-function on its corresponding interval of the domain.

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Domain of a Piecewise Function

The set of all possible input values for a piecewise function, considering the intervals of each sub-function.

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Range of a Piecewise Function

The set of all possible output values for a piecewise function, considering the outputs of each sub-function.

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Absolute Value Function

A function that outputs the distance of a number from zero, always a non-negative value.

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Breakpoint

The specific value in the domain where the rule of a piecewise function changes.

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Rewrite |x - 3| as a piecewise function

For x ≥ 3, the expression |x - 3| simplifies to x - 3. For x < 3, it becomes 3 - x.

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Rewrite |x| as a piecewise function

For x ≥ 0, the expression |x| equals x. For x < 0, it becomes -x.

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Rewrite |-x| as a piecewise function

For x ≥ 0, the expression |-x| equals -x. For x < 0, it becomes x.

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Rewrite |x + 5| as a piecewise function

For x ≥ -5, the expression |x + 5| simplifies to x + 5. For x < -5, it becomes -x - 5.

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Study Notes

Chapter 1 Fundamentals: Real Numbers

  • Objectives:
    • Real Numbers
    • Properties of Real Numbers
    • Addition and Subtraction
    • Multiplication and Division
    • The Real Line
    • Sets and Intervals
    • Absolute Value and Distance

Vocabulary

  • Natural Numbers: ..., -3, -2, -1, 0, 1, 2, 3,...
  • Whole Numbers: 0, 1, 2, 3, ...
  • Integers: ..., -3, -2, -1, 0, 1, 2, 3,...
  • Rational Numbers: Numbers that can be expressed as a fraction (p/q), where p and q are integers and q ≠ 0. Examples include fractions (1/2, 3/4), decimals with terminating or repeating digits (0.5, 0.333...), and integers.
  • Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).
  • Real Numbers: The set of all rational and irrational numbers. Represented by R. Examples include √2, π, 1/2, 1, etc.

Sets and Intervals

  • Set-builder notation: A concise way of defining a set by specifying its properties. Examples:

    • A = {x | x is an integer and 0 < x <7}
    • A is the set of all x such that x is an integer and 0 < x < 7.

  • Interval notation: A shorthand notation for representing intervals of real numbers. Examples:

    • (0, 7) (open interval)
    • [0, 7] (closed interval)
    • (-∞, ∞) (set of all real numbers)
    • [0, 7) (a closed and open interval)
    • (0, 7) (an open interval)

  • Union and Intersection: For sets S and T,

    • S∪T is the union of S and T (all elements in S or T).
    • S∩T is the intersection of S and T (all elements in both S and T).

  • Empty Set: Denoted by Ø, a set that contains no elements.

Absolute Value

  • Definition: |a| represents the distance from 'a' to zero on the number line.
    • If a ≥ 0, then |a| = a.
    • If a < 0, then |a| = -a.

Distance Between Points

  • Distance between points 'a' and 'b' on the real line: d(a, b) = |b−a|
  • The distance from a to b is the same as the distance from b to a.

Piecewise Functions

  • A function defined by multiple sub-functions, each applying to a specific part of the domain.
  • Each sub-function is associated with a specific interval or condition within the domain.

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Description

Explore the fundamentals of real numbers in this quiz covering their properties, operations, and the real line. This chapter also includes definitions and examples of natural, whole, integer, rational, and irrational numbers. Test your understanding of sets and intervals with key concepts presented here.

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