Numbers and Functions

1. What is a Number?

Calculus is built on the concept of real numbers and functions of real variables. Understanding the types of numbers and their properties is foundational for further study.

• Positive Integers: The simplest numbers, e.g., 1, 2, 3, ...

• Zero: The integer 0.

• Negative Integers: ..., -3, -2, -1

• Rational Numbers: Numbers that can be written as a fraction of two integers, e.g., , , (since ).

• Irrational Numbers: Numbers that cannot be written as a fraction of two integers, e.g., , .

• Real Numbers: All rational and irrational numbers together. Real numbers can be represented by infinite decimal expansions.

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

• Example: (repeating)

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

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

2. Intervals and Set Notation

Intervals are used to describe sets of real numbers between two endpoints.

• Open Interval:

• Closed Interval:

• Half-Open Intervals: or

Set Notation: Used to describe collections of numbers. For example, is the set of all real numbers whose square is less than 2.

3. Functions

A function is a rule which assigns to each input (from a set called the domain) exactly one output (in the range).

• Definition: A function from a set to a set is a rule that assigns to each in $D$ exactly one element in $R$.

• Domain: The set of all possible inputs for which the function is defined.

• Range: The set of all possible outputs.

Example: has domain (all real numbers), and range (all non-negative real numbers).

3.1. Graphing a Function

The graph of a function is the set of all points in the plane, where is in the domain of .

• 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.

3.2. Linear Functions

Linear functions are of the form , where is the slope and is the y-intercept.

• Slope: for two points and on the line.

• Graph: A straight line in the plane.

3.3. Domain and Range (Finding)

• To find the domain of a function given by a formula, determine all for which the formula makes sense (e.g., denominator not zero, square root of non-negative number).

• To find the range, solve for in terms of and determine all possible values $y$ can take.

Example: For , the domain is (all real numbers except 0), and the range is also $\mathbb{R} \setminus \{0\}$.

3.4. Functions in Real Life

Functions are used to describe relationships in science, engineering, economics, and everyday life. For example, the distance an object travels as a function of time, or the cost of goods as a function of quantity purchased.

4. Additional Key Concepts

• Piecewise Functions: Functions defined by different formulas on different intervals.

• Composite Functions: Functions formed by applying one function to the result of another, e.g., .

5. Example Table: Types of Numbers

Additional info: These notes summarize the foundational concepts from the first chapter of a standard college Calculus I course, including types of numbers, intervals, set notation, and the definition and properties of functions. These concepts are essential for understanding limits, derivatives, and integrals in later chapters.