Numbers and Functions

What is a Number?

Calculus is fundamentally concerned with functions of real variables, so understanding the types of numbers used is essential. Numbers can be classified into several categories:

• 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

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

Example: (repeating), (non-repeating)

Intervals and the Real Number Line

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

• Closed interval: includes endpoints and

• Open interval: excludes endpoints

• Half-open interval: or includes one endpoint

Distance on the real line: The distance between two numbers and is .

Set Notation

Sets are collections of numbers. Common notations include:

• : set of all real numbers

• : set of all rational numbers

• : set of all integers

• : intersection of sets and

• : union of sets and

Example: is the set of all such that .

Functions

Definition of a Function

A function assigns to each element in a set called the domain a unique element in another set called the range. The rule specifying the function is often given by a formula.

• Domain: Set of all input values for which the function is defined

• Range: Set of all possible output values

Example: has domain and range .

Graphing a Function

The graph of a function is the set of all points in the plane. The graph visually represents the relationship between input and output values.

• 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:

• is the slope

• is the y-intercept

The graph is a straight line. The slope measures the rate of change of with respect to .

Example: has slope $2.

Domain and Range from Formulas

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

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 on the real line is .

Example: The height of an object as a function of time, , models its motion.

Piecewise Defined Functions

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

Example:

Summary Table: Types of Numbers

Additional info:

• These notes cover foundational concepts for Calculus, including numbers, sets, intervals, and functions, which are essential for understanding limits, derivatives, and integrals in later chapters.