Numbers and Functions

1.1 What is a Number?

Calculus is fundamentally concerned with functions of real numbers. Understanding the types of numbers used in calculus is essential for further study.

• Positive Integers:

• Negative Integers:

• Zero: $0$

• Rational Numbers: Numbers that can be written as , where and are integers and .

• Irrational Numbers: Numbers that cannot be written as a ratio of integers (e.g., , ).

• Real Numbers: The set of all rational and irrational numbers. Every point on the number line corresponds to a real number.

Decimal Expansions: Rational numbers have either terminating or repeating decimal expansions. Irrational numbers have non-terminating, non-repeating decimals.

Example: ,

1.2 The Real Number Line and Intervals

The real number line is a geometric representation of all real numbers. Intervals are used to describe subsets of the real line.

• Closed Interval: includes both endpoints and .

• Open Interval: excludes both endpoints.

• Half-Open Intervals: or include only one endpoint.

• Distance on the Real Line: The distance between and is .

Example: The interval contains all real numbers such that .

1.3 Set Notation

Sets are collections of objects, often numbers. Set notation is used to describe intervals and other collections.

• Set-builder notation: denotes all such that .

• Union: is the set of elements in or .

• Intersection: is the set of elements in both and .

Example: , , .

1.4 Functions

A function is a rule that assigns to each element in a set (the domain) exactly one element (the value ) in another set (the range).

• Domain: The set of all for which is defined.

• Range: The set of all possible values can take.

• Notation: means is a function with domain and range .

Example: has domain and range .

1.5 Graphing a Function

The graph of a function is the set of all points in the plane, where is in the domain of .

• To graph , plot points for various and connect them smoothly if is continuous.

• The vertical line test determines if a curve is the graph of a function: any vertical line should intersect the graph at most once.

Example: The graph of is a parabola opening upwards.

1.6 Linear Functions

A linear function has the form , where is the slope and is the y-intercept.

• Slope: for two points and on the line.

• Linear functions graph as straight lines.

Example: has slope $2.

1.7 Domain and Range (Finding from Formulas)

To determine the domain of a function given by a formula, identify all for which the formula makes sense (e.g., avoid division by zero, square roots of negative numbers).

• Example: has domain .

• Example: has domain .

The range is found by considering all possible outputs as varies over the domain.

1.8 Functions in Real Life

Functions are used to model relationships in science, engineering, and everyday life. For example, the distance traveled by an object moving at constant speed for time is .

• Example: The height of a ball thrown upward as a function of time can be modeled by .

1.9 Piecewise Defined Functions

Some functions are defined by different formulas on different intervals. These are called piecewise functions.

• Example:

1.10 Exercises and Practice

Practice problems may include:

• Identifying types of numbers (integer, rational, irrational, real).

• Writing intervals in set notation.

• Finding the domain and range of given functions.

• Graphing basic functions and applying the vertical line test.

Table: Types of Numbers and Their Properties

Additional info: These notes are based on the first chapter of a first-semester calculus course, covering foundational concepts necessary for understanding limits, derivatives, and integrals in later chapters.