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 integer 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, , where .

• 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 by infinite decimal expansions.

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

Why are Real Numbers Called 'Real'?

• Historically, real numbers were distinguished from 'imaginary' numbers (involving ).

• Real numbers correspond to points on the number line, making them suitable for measuring continuous quantities.

Intervals and the Number Line

• Interval Notation: Used to describe sets of real numbers between two endpoints.

• Examples:

  • Open interval:

  • Closed interval:

  • Half-open interval:

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

2. Set Notation

Set notation is a concise way to describe collections of numbers, which is essential for defining domains and ranges of functions.

• Set-builder notation:

• Examples:

3. Functions

Functions are the central objects of study in calculus. A function assigns to each input exactly one output, according to a specific rule.

3.1. Definition of a Function

• A function is a rule that assigns to each element in a set (the domain) exactly one element in another set (the range).

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

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

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

3.2. Graphing a Function

• The graph of a function is the set of all points in the plane.

• To graph a function, plot points for various values and connect them smoothly if the function is continuous.

Example: The graph of is a parabola opening upwards.

3.3. Linear Functions

• A linear function has the form , where is the slope and is the y-intercept.

• The graph is a straight line.

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

3.4. Domain and 'Biggest Possible Domain'

• When defining a function by a formula, the domain is usually the set of all real numbers 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).

3.5. Example – Finding Domain and Range

• For , the domain is because you cannot take the square root of a negative number (in real numbers).

• The range is also .

3.6. Functions in Real Life

• Functions can model real-world relationships, such as distance as a function of time, or cost as a function of quantity.

• Example: The distance from a fixed point as a function of time if moving at constant speed is .

3.7. The 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.

• This test helps distinguish functions from relations that are not functions.

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

4. Summary Table: Types of Numbers

Additional info: This table summarizes the main types of numbers relevant to calculus, as described in the text.

5. Key Formulas and Properties

• Distance on the number line:

• Linear function:

• Domain of :

• Domain of :

6. Practice and Application

• Identify the domain and range of given functions.

• Use set notation to describe intervals and solution sets.

• Apply the vertical line test to determine if a graph represents a function.