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 is foundational for further study in calculus.

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

• Zero: The number 0.

• Negative Integers: ..., -3, -2, -1

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

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

• Real Numbers: All rational and irrational numbers together.

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

• Example: (repeating)

• Example: (non-repeating)

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. Set notation is a concise way to describe collections of numbers.

• Open Interval: is the set of all such that .

• Closed Interval: is the set of all such that .

• Half-Open Interval: or

• Set Notation:

Example: is the set of all between 1 and 6.

3. Functions

A function is a rule that assigns to each input (from the domain) exactly one output (in the range). Functions are central objects in calculus.

• Definition: A function from a set to a set assigns to each in $A$ a unique in $B$.

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

• Range: The set of all possible outputs.

• Formula: Functions are often given by formulas, e.g., .

Example: The function has domain and range .

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 functions of the form , where and are constants.

• Slope (): Measures the rate of change of the function.

• Intercept (): The value of the function when .

Example: is a linear function with slope 2 and intercept 3.

3.3. Domain and Range from Formula

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 .

3.4. Functions in Real Life

Functions can describe relationships in physics, economics, biology, and more. For example, the distance between two points on a line can be described by a function .

3.5. Piecewise Functions

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

• Example:

4. Exercises and Applications

• Find the 200th digit after the period in the decimal expansion of .

• Which of the following fractions have finite decimal expansions?

• Draw the following sets of real numbers.

• Suppose and are intervals. Is it always true that is an interval? What about ?

• Consider the set . Are these sets intervals?

5. Summary Table: Types of Numbers

Additional info: These notes cover foundational concepts for calculus, including types of numbers, intervals, set notation, and the definition and properties of functions. These are essential for understanding limits, derivatives, and integrals in later chapters.