Numbers and Functions

1.1 Different Kinds of Numbers

Calculus relies on a solid understanding of different types of numbers. The most basic are the positive integers (1, 2, 3, ...), the number zero (0), and the negative integers (..., -3, -2, -1). Together, these form the set of integers.

• Rational numbers: Numbers that can be written as a fraction of two integers, e.g., $\frac{1}{2}$, $-\frac{3}{4}$.

• Irrational numbers: Numbers that cannot be written as a fraction, e.g., $\sqrt{2}$, $\pi$.

• Real numbers: The set of all rational and irrational numbers.

Decimal expansions can be finite (e.g., 0.5), repeating (e.g., 0.333...), or non-repeating (e.g., $\sqrt{2}$).

1.2 A Reason to Believe in $\sqrt{2}$

The Pythagorean theorem shows that the diagonal of a unit square has length $\sqrt{2}$, which is not a rational number. This motivates the need for irrational numbers and the completeness of the real number system.

1.3 The Real Line and Intervals

The real line is a geometric representation of all real numbers as points on a line. Intervals are subsets of the real line, such as:

• Open interval: $(a, b)$ contains all $x$ such that $a < x < b$.

• Closed interval: $[a, b]$ contains all $x$ such that $a \leq x \leq b$.

• Half-open intervals: $[a, b)$ or $(a, b]$.

The distance between two numbers $a$ and $b$ is $|a - b|$.

1.4 Set Notation

Sets are collections of numbers. Common notations include:

• $\mathbb{R}$: set of all real numbers

• $\mathbb{Q}$: set of all rational numbers

• $\mathbb{Z}$: set of all integers

• $\mathbb{N}$: set of all positive integers

Example: $A = \{x \mid 0 < x < 1\}$ is the set of all $x$ between 0 and 1.

1.5 Exercises

• Determine if a decimal expansion is finite or infinite.

• Classify numbers as rational or irrational.

• Use set notation to describe intervals and sets.

Functions

3.1 Definition of a Function

A function $f$ assigns to each input $x$ in its domain exactly one output $f(x)$. The range is the set of all possible outputs.

• Domain: The set of all $x$ for which $f(x)$ is defined.

• Range: The set of all $f(x)$ values as $x$ varies over the domain.

3.2 Graphing a Function

The graph of a function is the set of all points $(x, f(x))$ in the plane. The vertical line test determines if a curve is the graph of a function: if any vertical line crosses the curve more than once, it is not a function.

3.3 Linear Functions

A linear function has the form:

• $f(x) = mx + b$

where $m$ is the slope and $b$ is the y-intercept. The graph is a straight line.

3.4 Domain and 'Biggest Possible Domain'

When given a formula, the domain is all real numbers for which the formula makes sense (e.g., no division by zero, no square roots of negative numbers).

3.5 Example: Finding Domain and Range

For $f(x) = 1/\sqrt{x}$, the domain is $x > 0$ (since the square root and division by zero are undefined for $x \leq 0$).

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.

3.7 The Vertical Line Property

This property is used to determine if a graph represents a function. If every vertical line crosses the graph at most once, the graph is a function.

Table: Types of Numbers

Summary: Real numbers include all rational and irrational numbers, and functions are rules that assign each input in the domain to exactly one output in the range. Understanding these foundational concepts is essential for studying calculus.