Numbers and Functions

1. What is a Number?

Calculus begins with understanding the types of numbers used in mathematics, especially real numbers. This section introduces the basic kinds of numbers and their properties.

• Positive Integers: The simplest numbers, such as 1, 2, 3, ...

• Zero: The integer 0, which is neither positive nor negative.

• Negative Integers: Numbers like -1, -2, -3, ...

• Rational Numbers: Numbers that can be written as a fraction of two integers, , where .

• Irrational Numbers: Numbers that cannot be written as a fraction, such as or .

• Real Numbers: All rational and irrational numbers together form the set of real numbers, .

Decimal Expansions: Rational numbers have either terminating or repeating decimals, while irrational numbers have non-repeating, non-terminating decimals.

• Example: (repeating)

• Example: (non-repeating)

Distance on the Number Line: The distance between two numbers and is .

2. Intervals and Set Notation

Intervals are used to describe subsets of real numbers. Set notation helps specify these intervals and other collections of numbers.

• Open Interval: contains all such that .

• Closed Interval: contains all such that .

• Half-Open Interval: or .

• Set Notation:

Example: is the set of all such that .

3. Functions

A function is a rule that assigns to each input (from its domain) exactly one output (in its range). Functions are central to calculus and are often written as .

• Definition: To specify a function , you must define its domain and the rule for computing .

• Domain: The set of all possible input values for which the function is defined.

• Range: The set of all possible output values.

• Linear Functions: Functions of the form , where is the slope and is the y-intercept.

Example: is a linear function with slope 2 and y-intercept 3.

4. Graphing Functions

The graph of a function is a visual representation of all pairs for in the domain. Graphs help understand the behavior of functions.

• Vertical Line Test: A curve is the graph of a function if and only if no vertical line intersects the curve more than once.

• Domain and Range from Graphs: The domain is the set of -values for which the graph exists; the range is the set of -values the graph attains.

Example: The graph of is a parabola; its domain is and its range is .

5. Piecewise Functions

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

• Example:

6. Applications and Real-Life Examples

Functions are used to model real-life situations, such as distance, speed, and growth. Understanding domain and range is crucial for interpreting these models.

• Distance Function: could represent the distance traveled at time .

• Example: The function models the side length of a square given its area .

7. Summary Table: Types of Numbers

8. Key Formulas and Properties

• Distance on the Real Line:

• Linear Function:

• Piecewise Function Example:

Additional info: These notes cover foundational concepts for calculus, including types of numbers, set notation, functions, and graphing. Later chapters (as seen in the table of contents) will address limits, derivatives, integrals, and applications, which are core topics in a college calculus course.