Numbers and Functions

1.1 What is a Number?

Calculus relies on a solid understanding of different types of numbers and their properties. The simplest numbers are the positive integers (1, 2, 3, ...), which are extended to include zero and the negative integers (..., -3, -2, -1, 0, 1, 2, 3, ...).

• Rational numbers are numbers that can be written as a ratio of two integers: , where .

• Irrational numbers cannot be written as a ratio of integers (e.g., , ).

• Real numbers include both rational and irrational numbers, forming a continuous number line.

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.

Example: ,

1.2 The Real Number Line and Intervals

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

• Closed interval:

• Open interval:

• Half-open intervals: or

The distance between two real numbers and is .

Example: The interval includes all real numbers from 0 to 1, including the endpoints.

1.3 Set Notation

Sets are collections of numbers or objects. Common notations include:

• : the set of all real numbers

• : the set of all rational numbers

• : the set of all integers

• : the intersection of sets A and B (elements in both)

• : the union of sets A and B (elements in either)

Example: , ,

1.4 Functions

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. We write .

• Domain: The set of all input values for which the function is defined.

• Range: The set of all possible output values .

Example: has domain and range .

1.5 Graphing a Function

The graph of a function is the set of all points in the plane. The vertical line test is used to determine if a graph represents a function: if any vertical line intersects the graph at more than one point, it is not a function.

Example: The graph of passes the vertical line test, but the graph of a circle does not.

1.6 Linear Functions

A linear function has the form , where is the slope and is the y-intercept. The graph is a straight line.

• Slope:

• Y-intercept: The value of when

Example: has slope 2 and y-intercept 1.

1.7 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., avoid division by zero, square roots of negative numbers).

Example: has domain .

1.8 Functions in Real Life

Functions are used to model relationships in science, engineering, and everyday life. For example, the distance traveled by an object moving at constant speed for time is .

1.9 Piecewise Defined Functions

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

Example:

1.10 The Vertical Line Property

The vertical line property states that a curve in the plane is the graph of a function if and only if no vertical line intersects the curve more than once.

Table: Types of Numbers

Summary

• Understanding numbers and their properties is foundational for calculus.

• Functions describe relationships between variables and are represented graphically.

• Domains and ranges must be carefully considered when working with functions.

• Set notation and interval notation are essential tools for describing mathematical objects.