Numbers and Functions

What is a Number?

Calculus begins with the study of functions of real variables, which requires a clear understanding of what numbers are. Numbers are classified into several types, each with distinct properties and uses in mathematics.

• Positive Integers:

• Negative Integers:

• Zero: $0$

• Rational Numbers: Numbers that can be expressed as a ratio of two integers, , where .

• Real Numbers: All numbers that can be represented on the number line, including both rational and irrational numbers.

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

Additional info: Real numbers include both rational and irrational numbers, and every point on the number line corresponds to a real number.

Intervals and the Number Line

Intervals are used to describe sets of real numbers between two endpoints. The distance between two numbers and is .

• Closed Interval: includes both endpoints.

• Open Interval: excludes both endpoints.

• Half-Open Interval: or includes one endpoint.

Example: The interval includes all real numbers such that .

Set Notation

Sets are collections of numbers or objects. In calculus, sets are often used to describe domains and ranges of functions.

• Set of all such that :

• Union of sets and :

• Intersection of sets and :

Example: , ,

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 () that the function can produce.

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 . The graph visually represents the relationship between $x$ and .

• 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:

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

• Intercept (): The value of when .

Example: The graph of is a straight line with slope $2y.

Domain and Range from Formulas

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

• Example: For , the domain is all real numbers except .

• Example: For , the domain is .

Functions in Real Life

Functions are used to model relationships in science, engineering, and everyday life. For example, the distance between two points on a number line is a function of their coordinates.

• Distance Function:

Example: The function gives the distance between two points and on the real number line.

Piecewise Defined Functions

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

Example: The absolute value function:

Summary Table: Types of Numbers

Additional info:

These notes cover foundational concepts in calculus, including numbers, sets, functions, domains, ranges, and graphing. Understanding these basics is essential for progressing to limits, derivatives, and integrals.