Numbers and Functions

1. What is a Number?

The concept of a number is foundational in calculus and mathematics as a whole. Numbers are used to count, measure, and label. In calculus, we primarily work with real numbers, which include several important subsets.

• Positive integers: $1, 2, 3, \ldots$

• Negative integers: $-1, -2, -3, \ldots$

• Zero: $0$

• Rational numbers: Numbers that can be written as a fraction $\frac{m}{n}$, where $m$ and $n$ are integers and $n \neq 0$.

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

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

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

Example: $\frac{1}{3} = 0.333\ldots$, $\sqrt{2} = 1.4142135\ldots$

Why are real numbers called 'real'? The term distinguishes them from 'imaginary' numbers, which involve the square root of negative numbers. Real numbers can be represented on the number line.

2. Intervals and the Number Line

Intervals are used to describe sets of real numbers between two endpoints. They are fundamental in calculus for specifying domains and ranges.

• Closed interval: $[a, b]$ includes both endpoints $a$ and $b$.

• Open interval: $(a, b)$ excludes both endpoints.

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

Distance on the number line: The distance between two numbers $x$ and $y$ is $|x - y|$.

3. Set Notation

Set notation is a concise way to describe collections of numbers. It is widely used in calculus to define domains, ranges, and solution sets.

• Example: $A = \{ x \mid 1 < x < 6 \}$ is the set of all $x$ such that $1 < x < 6$.

• Union: $A \cup B$ is the set of elements in $A$ or $B$.

• Intersection: $A \cap B$ is the set of elements in both $A$ and $B$.

4. Functions

Functions are central objects in calculus. A function assigns to each input exactly one output.

• Definition: A function $f$ from a set $A$ to a set $B$ is a rule that assigns to each element $x$ in $A$ exactly one element $f(x)$ in $B$.

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

• Range: The set of all possible outputs.

Example: $f(x) = x^2$ has domain $\mathbb{R}$ (all real numbers) and range $[0, \infty)$ (all non-negative real numbers).

4.1. Graphing a Function

The graph of a function is the set of all points $(x, f(x))$ in the plane. It visually represents the relationship between input and output.

• 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.

4.2. Linear Functions

Linear functions are of the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Their graphs are straight lines.

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Equation of a line: $y = mx + b$

4.3. Domain and Range

To find the domain of a function, determine all $x$-values for which the formula makes sense (e.g., avoid division by zero or taking square roots of negative numbers).

Example: For $f(x) = \frac{1}{x}$, the domain is $\mathbb{R} \setminus \{0\}$ (all real numbers except $0$).

4.4. Functions in Real Life

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

5. Tables and Visuals

The following table summarizes types of numbers and examples:

Additional info: Table entries inferred and summarized for clarity.

6. Key Formulas and Properties

• Distance on the number line: $|x - y|$

• Equation of a line: $y = mx + b$

• Function notation: $f: A \to B$, $f(x)$

7. Examples and Applications

• Example 1: The function $f(x) = \sqrt{x}$ has domain $[0, \infty)$ and range $[0, \infty)$.

• Example 2: The function $f(x) = \frac{1}{x}$ has domain $\mathbb{R} \setminus \{0\}$ and range $\mathbb{R} \setminus \{0\}$.

• Application: Modeling the height of a falling object as a function of time.