Numbers and Functions

What is a Number?

Calculus begins with understanding the types of numbers used in mathematics, especially real numbers. These form the basis for functions and further concepts in calculus.

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

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

• Example: (repeating), (non-repeating)

The Real Number Line and Intervals

The real number line is a visual representation of all real numbers. Intervals are subsets of the real line, defined by their endpoints.

• Closed Interval: includes both endpoints and .

• Open Interval: excludes both endpoints.

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

Set Notation

Sets are collections of numbers. Interval notation and set-builder notation are used to describe sets of real numbers.

• Example: is 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 is given by the formula:

where is the slope and is the y-intercept. The graph is a straight line.

• Slope:

• 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 are used to model relationships in science, engineering, and everyday life. For example, the distance from a point to the origin is a function of its coordinates:

Piecewise Defined Functions

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

• Example:

Summary Table: Types of Numbers

Additional info:

• These notes cover foundational concepts for calculus, including numbers, sets, functions, domains, ranges, and graphing. Later chapters (as seen in the table of contents) will cover limits, derivatives, integrals, and applications, which are core calculus topics.