Numbers and Functions

What is a Number?

Calculus begins with the study of functions of real variables, which requires understanding the types of numbers used in mathematics.

• Positive Integers:

• Negative Integers:

• Zero: $0$

• Rational Numbers: Numbers that can be written as , where and are integers and .

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

Example: are rational numbers; are irrational numbers.

Decimal Expansions and Real Numbers

Every real number can be represented by a (possibly infinite) decimal expansion. Rational numbers have repeating or terminating decimals, while irrational numbers have non-repeating, non-terminating decimals.

• Finite Decimal:

• Repeating Decimal:

• Irrational Decimal:

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

The Real Number Line and Intervals

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

• Closed Interval: includes endpoints and .

• Open Interval: excludes endpoints.

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

Example: The interval contains all real numbers between $0, including $0.

Set Notation

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

• Set of all such that :

• Union: is the set of elements in or .

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

• Range: The 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 and .

• To graph , plot points for various and connect them smoothly.

• The vertical line test determines if a graph represents a function: a vertical line should intersect the graph at most once.

Example: The graph of is a parabola; the graph of is defined only for .

Linear Functions

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

• Slope:

• Graph: A straight line in the plane.

Example: is a line with 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 moving object to a fixed point can be a function of time .

• Example: for constant speed .

The Vertical Line Property

A graph represents a function if and only if no vertical line intersects the graph more than once. This ensures each input has a unique output .

• Example: The graph of a circle fails the vertical line test and is not a function.

Tables: Types of Numbers

Summary

• Numbers are classified as integers, rational, and real numbers.

• Functions assign each input a unique output and are represented by graphs.

• Domains and ranges are essential for understanding where functions are defined and what values they can take.

• Linear functions are the simplest type, with straight-line graphs.

• The vertical line test helps identify valid functions from their graphs.