Numbers and Functions

What is a Number?

Calculus begins with understanding the types of numbers used in mathematics, especially real numbers. This section introduces the basic kinds of numbers and their properties.

• 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.

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

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

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

Intervals and Set Notation

Intervals are used to describe sets of real numbers between two endpoints. Set notation is essential for specifying domains and ranges in calculus.

• Closed Interval: includes both endpoints and .

• Open Interval: excludes both endpoints.

• Half-Open Interval: or includes one endpoint.

• Set Notation: means the set of all such that .

Example: The set is all real numbers between 1 and 6, not including 1 or 6.

Functions

Functions are central to calculus, describing how one quantity depends on another. A function assigns to each input exactly one output.

• 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.

Example: has domain and range .

Graphing a Function

The graph of a function is a visual representation showing the relationship between input and output values.

• Vertical Line Test: A curve is the graph of a function if and only if no vertical line intersects the curve more than once.

• Linear Functions: is a straight line with slope and -intercept .

Example: The graph of is a parabola; the graph of is a straight line.

Domain and Range from Formulas

To find the domain and range of a function given by a formula, determine for which inputs 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 .

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 described as a function of time.

• Example: The distance between two points and in the plane is .

Summary Table: Types of Numbers

Additional info:

• Set notation and interval notation are foundational for specifying domains and ranges in calculus problems.

• Understanding the types of numbers and their properties is essential for grasping more advanced calculus concepts such as limits, continuity, and differentiability.