Numbers and Functions

1.1 What is a Number?

Calculus begins with understanding the types of numbers used in mathematics, especially real numbers. This section introduces the different kinds of numbers and their properties.

• Positive Integers:

• Negative Integers:

• Zero: $0$

• Rational Numbers: Numbers that can be written as , where and are integers and .

• Irrational Numbers: Numbers that cannot be written as a ratio of two integers, e.g., , .

• Real Numbers: All rational and irrational numbers; can be represented on the number line.

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

1.2 The Real Number Line and Intervals

The real number line is a visual representation of all real numbers. Intervals are subsets of the real number line defined by endpoints.

• Closed Interval: includes both endpoints and .

• Open Interval: excludes both endpoints.

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

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

1.3 Set Notation

Sets are collections of numbers or objects. In calculus, sets are used to describe domains, ranges, and intervals.

• Set of all such that :

• Union of Sets: is the set of elements in or .

• Intersection of Sets: is the set of elements in both and .

Example: , ,

1.4 Functions

A function is a rule that assigns to each input (from the domain) exactly one output (in the range). Functions are central to calculus.

• Definition: A function from a set to a set assigns to each in $A$ a unique in $B$.

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

• Range: The set of all possible output values.

• Formula: Functions are often given by formulas, e.g., .

Example: has domain and range .

1.5 Graphing a Function

The graph of a function is a visual representation showing the relationship between input and output values.

• Graph: The set of points for in the domain.

• Vertical Line Test: A curve is the graph of a function if no vertical line intersects it more than once.

Example: The graph of is a parabola. The graph of a circle fails the vertical line test and is not a function.

1.6 Linear Functions

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

• Slope:

• Graph: A straight line in the plane.

Example: is a linear function with slope $2.

1.7 Domain and Range from Formulas

To find the domain and range of a function given by a formula, determine the set of input values for which the formula makes sense, and the corresponding output values.

• Domain: All for which is defined.

• Range: All possible values of .

Example: For , the domain is , and the range is $\mathbb{R} \setminus \{0\}$.

1.8 Functions in Real Life

Functions are used to model relationships in science, engineering, and everyday life. For example, the distance between two points, the speed of an object, or the cost of goods can all be represented as functions.

• Distance Function:

• Speed Function:

Example: The height of a ball thrown upward as a function of time.

1.9 Exercises and Practice Problems

Practice problems help reinforce understanding of numbers, sets, and functions.

• Find the 200th digit after the period in the decimal expansion of .

• Which of the following fractions have finite decimal expansions?

• Given sets and , find and .

• Determine the domain and range of .

HTML Table: Types of Numbers

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and applications have been added to illustrate key concepts.