Numbers and Functions

What is a Number?

Calculus begins with the study of functions of one real variable, which requires understanding the nature of numbers and functions. 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 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 together.

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

• Example: (repeating)

• Example: (non-repeating)

Visualization: Real numbers can be represented on a number line, with the distance between two numbers and given by .

Intervals and Set Notation

Intervals are used to describe sets of real numbers between two endpoints. Set notation is a concise way to specify collections of numbers.

• Closed Interval: includes both endpoints and .

• Open Interval: excludes both endpoints.

• Half-Open Interval: or includes one endpoint.

• Set Notation: means the set of all such that .

Example: The set is all real numbers between 1 and 6.

Functions

A function is a rule that assigns to each element in a set called the domain a unique element in a set called the range. Functions are central objects in calculus.

• 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 is defined.

• Range: The set of all possible output values .

Example: has domain (all real numbers) and range (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 how changes as $x$ varies.

• Linear Functions: is a straight line with slope and -intercept .

• 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.

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 (all real numbers except 0).

• Example: has domain .

Functions in Real Life

Functions model relationships in science and engineering, such as distance, velocity, and growth. For example, the distance from a point to the origin is a function .

• Example: The height of an object as a function of time .

• Example: The temperature at a location as a function of time .

Summary Table: Types of Numbers

Additional info:

• Piecewise functions are functions defined by different formulas on different intervals.

• Functions are foundational for all further topics in calculus, including limits, derivatives, and integrals.