Numbers and Functions

1. What is a Number?

Calculus is built on the concept of functions of one real variable. To understand this, we first explore the nature of numbers, especially real numbers, and their properties.

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

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

• Negative Integers: Numbers like -1, -2, -3, ...

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

• Decimal Expansions: Rational numbers have either finite or repeating decimal expansions, e.g.,

• Irrational Numbers: Numbers that cannot be written as a fraction, such as or . Their decimal expansions are infinite and non-repeating.

Example: (non-repeating, infinite decimal expansion)

2. Real Numbers and Intervals

Real numbers include all rational and irrational numbers. They can be visualized as points on a continuous number line.

• Intervals: A set of real numbers between two endpoints. For example, the interval includes all such that .

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

Example: The interval includes all real numbers such that .

3. Set Notation

Sets are collections of numbers. Set notation is used to describe intervals and other collections of numbers.

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

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

Example: , ,

4. Functions

A function assigns to each input (from its domain) exactly one output (in its range). Functions are central to calculus.

• Definition: A function is a rule that assigns to each in its domain a unique value .

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

• Range: The set of all possible output values the function can produce.

• Graph of a Function: The set of all points in the plane.

Example: For , the domain is all real numbers, and the range is all non-negative real numbers.

5. Linear Functions

Linear functions are functions of the form , where and are constants.

• Slope (): Measures the steepness of the line.

• Intercept (): The point where the line crosses the -axis.

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

6. Piecewise Functions

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

• Example:

7. Domain and Range: Examples

To find the domain and range of a function, analyze where the formula makes sense and what outputs are possible.

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

8. Functions in Real Life

Functions can describe relationships in physics, economics, biology, and more. For example, the distance between two points, the growth of a population, or the cost of production can all be modeled by functions.

• Example: The distance between two points and on a line is .

9. The Vertical Line Test

The vertical line test is a graphical method to determine if a curve is the graph of a function. If any vertical line crosses the graph more than once, it is not a function.

• Example: The graph of passes the vertical line test, but the graph of a circle does not.

10. Table: Types of Numbers

The following table summarizes the main types of numbers discussed:

Additional info: These notes cover foundational concepts for Calculus I, including numbers, intervals, set notation, functions, domains, ranges, and graphical representations. These are essential for understanding limits, derivatives, and integrals in later chapters.