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., $\frac{1}{2}$, $-\frac{3}{4}$, $5$ (since $5 = \frac{5}{1}$).

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

• Real Numbers: The set of all rational and irrational numbers. Every point on the number line corresponds to a real number.

Decimal Expansions: Rational numbers have decimal expansions that either terminate or repeat. Irrational numbers have non-terminating, non-repeating decimals.

• Example: $\frac{1}{3} = 0.333...$ (repeating)

• Example: $\sqrt{2} = 1.4142135...$ (non-repeating, non-terminating)

Intervals: An interval is a set of real numbers between two endpoints. For example, the interval $[a, b]$ includes all $x$ such that $a \leq x \leq b$.

2. Set Notation

Set notation is used to describe collections of numbers, especially intervals and solution sets.

• Interval Notation: $[a, b]$ (closed), $(a, b)$ (open), $[a, b)$ or $(a, b]$ (half-open).

• Set-builder Notation: $\{ x \mid a < x < b \}$ means the set of all $x$ such that $a < x < b$.

• Union and Intersection: $A \cup B$ (union), $A \cap B$ (intersection).

Example: The set $B = \{ x \mid 1 < x \leq 3 \}$ is the interval $(1, 3]$.

3. Functions

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

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

• Notation: $f: X \to Y$ means $f$ is a function from set $X$ (domain) to set $Y$ (codomain).

• Formula: Functions are often given by a formula, e.g., $f(x) = x^2 + 1$.

• Domain: The set of all $x$ for which the formula makes sense.

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

Example: For $f(x) = \sqrt{x}$, the domain is $x \geq 0$ (since square roots of negative numbers are not real), and the range is $f(x) \geq 0$.

3.1. Graphing a Function

The graph of a function is the set of all points $(x, f(x))$ in the plane, where $x$ is in the domain of $f$.

• 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 $y = x^2$ is a parabola. The graph of $x^2 + y^2 = 1$ (a circle) is not the graph of a function $y$ of $x$ because it fails the vertical line test.

3.2. Linear Functions

A linear function has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$ for two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line.

• Graph: The graph is a straight line.

Example: $f(x) = 2x + 3$ is a linear function with slope 2 and y-intercept 3.

3.3. Domain and Range

To find the domain of a function given by a formula, determine all $x$ for which the formula is defined (e.g., avoid division by zero, square roots of negative numbers).

To find the range, determine all possible values $f(x)$ can take as $x$ varies over the domain.

• Example: $f(x) = \frac{1}{x}$ has domain $x \neq 0$ and range $f(x) \neq 0$.

3.4. Functions in Real Life

Functions are used to model relationships in science, engineering, and everyday life. For example, the distance $d$ traveled by a car moving at constant speed $v$ for time $t$ is $d = vt$.

3.5. Piecewise Functions

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

• Example: $f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

4. Table: Types of Numbers

The following table summarizes the main types of numbers discussed:

5. Additional Info

• Distance on the Number Line: The distance between two numbers $a$ and $b$ is $|a - b|$.

• Half-open Intervals: $[a, b)$ includes $a$ but not $b$; $(a, b]$ includes $b$ but not $a$.

• Set Operations: The union $A \cup B$ contains all elements in $A$ or $B$; the intersection $A \cap B$ contains all elements in both $A$ and $B$.