Numbers and Functions

1. What is a Number?

This section introduces the foundational concept of numbers, which are essential for all of calculus. Understanding the types and properties of numbers is crucial for working with functions and calculus concepts.

• Positive Integers: The simplest numbers, such as 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., , , $5).

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

• Real Numbers: The set of all rational and irrational numbers. Every point on the number line corresponds to a real number.

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

Example: ,

2. The Real Number Line and Intervals

The real number line is a geometric representation of all real numbers. Intervals are subsets of the real line and are used to describe domains and ranges of functions.

• Closed Interval: includes all numbers such that .

• Open Interval: includes all numbers such that .

• Half-Open Intervals: or include one endpoint but not the other.

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

Example: The interval contains all such that .

3. Set Notation

Set notation is used to describe collections of numbers, such as intervals or solution sets.

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

• Union: is the set of all elements in or .

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

Example: , , then .

4. Functions

Functions are central objects in calculus, describing relationships between variables. A function assigns to each input exactly one output.

• Definition: A function from a set to a set assigns to each element in $A$ exactly one element in $B$.

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

• Range: The set of all possible output values.

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

Example: For , the domain is (since square roots of negative numbers are not real).

4.1. 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 is defined, and the range is the set of all possible $f(x)$ values.

• 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 opening upwards.

4.2. 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 .

• Example: has domain .

4.3. Functions in Real Life

Functions are used to describe relationships in science, engineering, economics, and everyday life. For example, the distance traveled at constant speed in time is .

5. Summary Table: Types of Numbers

Key Point: All integers and rational numbers are real numbers, but not all real numbers are rational.

6. Additional Info

• Notation: denotes the set of all real numbers, the set of all rational numbers, the set of all integers.

• Absolute Value: denotes the distance from to 0 on the real number line.