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 operations.

• 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., , , (since ).

• 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. Real numbers can be represented as points on the number line.

Decimal Expansions: Every real number can be represented by a (possibly infinite) decimal expansion. Rational numbers have repeating or terminating decimals, while irrational numbers have non-repeating, non-terminating decimals.

• Example: (repeating decimal)

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

Intervals: An interval is a set of real numbers between two endpoints. For example, the interval includes all such that .

2. Set Notation

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

• Interval Notation: denotes all with ; denotes all $x$ with .

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

• Union and Intersection: The union is the set of elements in or ; the intersection is the set of elements in both $A$ and $B$.

Example: The set is the set of all such that .

3. Functions

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

• Definition: A function from a set to a set is a rule that 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.

• Notation: means is a function from to .

Example: has domain (all real numbers), and range (all non-negative real numbers).

3.1. 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 -values that $f(x)$ can take.

• 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.

3.2. Linear Functions

Linear functions are functions of the form , where is the slope and is the -intercept.

• Slope: The rate of change of the function. For two points and , the slope is .

• Graph: The graph of a linear function is a straight line.

Example: has slope 2 and -intercept 1.

3.3. Domain and Range

To find the domain of a function given by a formula, determine all 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 model real-world relationships, such as distance over time, population growth, or temperature changes.

• Example: The distance traveled at constant speed over time is .

4. Summary Table: Types of Numbers

Additional info: These notes are based on the first chapter of a standard college Calculus I course, covering foundational concepts necessary for further study in calculus, such as types of numbers, set notation, and the definition and properties of functions.