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 main classes of numbers used in calculus and their significance.

• 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 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-terminating, non-repeating decimals.

Example: ,

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 defined by inequalities.

• Open Interval:

• Closed Interval:

• Half-Open Intervals: or

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

Example: The interval includes all real numbers between 0 and 1, including the endpoints.

3. Set Notation

Set notation is used to describe collections of numbers. Common notations include:

• : The set of all real numbers

• : The set of all rational numbers

• : The set of all integers

• : The set of elements in both and (intersection)

• : The set of elements in or (union)

Example: , , then .

4. Functions

Functions are fundamental objects in calculus, describing how one quantity depends on another.

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

• Domain: The set of all for which is defined.

• Range: The set of all possible values can take.

Example: has domain and range .

4.1. Graphing a Function

The graph of a function is the set of all points in the plane. It visually represents the relationship between and .

• Vertical Line Test: A curve 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. Linear Functions

Linear functions have the form , where is the slope and is the y-intercept.

• Slope:

• Graph: A straight line in the plane.

Example: is a line with slope 2 and y-intercept 1.

4.3. Piecewise Functions

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

Example:

4.4. Domain and Range (Finding)

To find the domain, determine all for which the formula makes sense (e.g., avoid division by zero, square roots of negative numbers). The range is all possible values.

Example: For , the domain is .

4.5. Functions in Real Life

Functions can model real-world relationships, such as distance over time, population growth, or physical laws.

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

4.6. The Vertical Line Property

This property is used to determine if a curve in the plane represents a function: if every vertical line crosses the curve at most once, the curve is the graph of a function.

5. Exercises and Applications

• Identify the type of number (integer, rational, irrational, real) for given examples.

• Find the domain and range of various functions.

• Graph simple functions and apply the vertical line test.

• Use set notation to describe intervals and solution sets.

6. Summary Table: Types of Numbers

Key Point: Real numbers include all numbers that can be represented on the number line, both rational and irrational.

7. Additional Info

• Notation: means is a function from set (domain) to set (codomain).

• Piecewise Functions: Useful for modeling situations where a rule changes based on input value.