Numbers and Functions

1. What is a Number?

The study of calculus begins with understanding the types of numbers used in mathematics and their properties. This section introduces the foundational number systems and their significance in calculus.

• Positive Integers: The simplest numbers, denoted as 1, 2, 3, ...

• Zero: The number 0, which is neither positive nor negative.

• Negative Integers: ..., -3, -2, -1

• Rational Numbers: Numbers that can be written as a fraction of two integers, , where .

• Irrational Numbers: Numbers that cannot be written as a simple fraction, such as or .

• Real Numbers: The set of all rational and irrational numbers. Real numbers can be represented by infinite decimal expansions.

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

2. The Real Number Line and Intervals

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

• Open Interval: contains all such that .

• Closed Interval: contains all such that .

• Half-Open Intervals: or

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

3. Set Notation

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

• Set-builder notation: denotes the set of all such that .

• Union: is the set of elements in or .

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

Example: is the set of all such that or .

4. Functions

A function is a rule that assigns to each element in a set (the domain) exactly one element (the value of the function at $x$).

• Definition: A function from a set to a set is a rule that assigns to each in $A$ a unique in $B$, denoted .

• Domain: The set of all for which is defined.

• Range: The set of all possible values can take.

Example: has domain (all real numbers) and range .

5. Graphing a Function

The graph of a function is the set of all points in the plane. The domain is the set of -values for which the function is defined, and the range is the set of -values the function attains.

• 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 is a parabola; the graph of is defined only for .

6. Domain and Range: Examples

To find the domain and range of a function, determine the set of -values 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 .

7. Functions in Real Life

Functions are used to model relationships in science, engineering, and everyday life. For example, the distance an object travels as a function of time, or the temperature as a function of location.

8. Summary Table: Types of Numbers

Additional info: The above notes are based on the first chapter of a standard college calculus course, covering foundational concepts necessary for further study in calculus, such as limits, derivatives, and integrals.