Numbers and Functions

Different Kinds of Numbers

Calculus begins with a study of numbers and functions, focusing on real numbers and their properties. Understanding the types of numbers is foundational for all further topics.

• 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 two integers, e.g., , .

• Real Numbers: All rational and irrational numbers; can be represented on the number line.

Example: (repeating decimal), (non-repeating, non-terminating decimal).

Decimal Expansions and Real Numbers

Real numbers can be represented by decimal expansions, which may be finite, repeating, or non-repeating.

• Finite Decimal:

• Repeating Decimal:

• Non-repeating Decimal:

Key Point: Every real number corresponds to a unique point on the number line.

Intervals and the Number Line

Intervals are used to describe sets of real numbers between two endpoints.

• Closed Interval: includes both endpoints.

• Open Interval: excludes both endpoints.

• Half-Open Interval: or includes one endpoint.

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

Set Notation

Sets are collections of numbers, often described using interval notation or set-builder notation.

• Set-builder notation: means the set of all such that .

• Union: is the set of elements in or .

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

Functions

Definition of a Function

A function is a rule that assigns to each element in a set called the domain exactly one element in a set called the range. The function is often written as .

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

• Range: The set of all possible output values ().

Example: For , the domain is all real numbers, and the range is all non-negative real numbers.

Graphing a Function

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

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

Example: The graph of passes the vertical line test, but the graph of a circle does not.

Linear Functions

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

• Slope (): Measures the rate of change; .

• Y-intercept (): The value of when .

Example: has slope $2.

Domain and Range from Formulas

To find the domain and range of a function given by a formula, determine for which the formula makes sense (e.g., no division by zero, no square roots of negative numbers).

• Example: has domain .

• Example: has domain .

Functions in Real Life

Functions model relationships in science, engineering, and everyday life. For example, the distance from a point to the origin is .

• Example: The height of a ball thrown upward as a function of time.

Piecewise Defined Functions

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

• Example:

Summary Table: Types of Numbers

Summary Table: Interval Notation

Additional info:

• These notes cover foundational concepts for Calculus, including numbers, functions, domains, ranges, and graphing. Later chapters (as seen in the table of contents) will address limits, derivatives, integrals, and their applications.