Numbers and Functions

1. What is a Number?

Calculus relies on a solid understanding of different types of numbers and their properties. This section introduces the foundational number systems used in calculus.

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

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

• 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 simple fraction, e.g., $$

• Real Numbers: All rational and irrational numbers together.

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

2. The Real Number Line and Intervals

The real number line is a geometric representation of all real numbers as points on a line. Intervals are subsets of the real line, often used to specify domains and ranges of functions.

• Open Interval: $$

• Closed Interval: $$

• Half-Open Intervals: $$

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

Example: The interval includes all real numbers between -1 and 2, including the endpoints.

3. Set Notation

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

• Set-builder notation: $xa < x < b$.

• Union: $AB$.

• Intersection: $AB$.

4. Functions

Functions are central objects in calculus, describing relationships between variables.

• Definition: A function assigns to each element in a set (the domain) exactly one element in another set (the range).

• Notation: $fXY$ (codomain).

• Example: $\mathbb{R}[0, \infty)$.

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 values.

• 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.

4.2. Linear Functions

Linear functions are the simplest type of functions, represented by straight lines.

• General form: $mby$-intercept.

• Graph: A straight line with slope and intercept .

4.3. Domain and Range

The domain of a function is the set of all input values for which the function is defined. The range is the set of all possible output values.

• Example: For $[0, \infty)[0, \infty)$.

4.4. Piecewise Functions

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

• Example: $$

4.5. Functions in Real Life

Functions are used to model relationships in science, engineering, economics, and everyday life. For example, the distance traveled as a function of time, or the cost as a function of quantity produced.

5. Tables and Visual Summaries

The following table summarizes the main types of numbers discussed:

Key Point: Real numbers include both rational and irrational numbers, and are represented as points on the real number line.

6. Additional Info

• Set notation is essential for describing domains, ranges, and solution sets in calculus.

• Functions can be represented algebraically, graphically, or numerically (e.g., in tables).

• Piecewise functions and vertical line test are important for understanding which curves represent functions.