Numbers and Functions

1. What is a Number?

Calculus is fundamentally concerned with functions of real variables. To understand this, we begin by exploring the concept of numbers, particularly real numbers, and their properties.

• Positive Integers: The simplest numbers, denoted as 1, 2, 3, ...

• Zero: The number 0.

• Negative Integers: ..., -3, -2, -1.

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

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

• Real Numbers: The set of all rational and irrational numbers. Real numbers can be represented by infinite decimal expansions.

Example: $\frac{1}{3} = 0.3333\ldots$ (repeating decimal), $\sqrt{2} = 1.4142135\ldots$ (non-repeating, non-terminating decimal).

2. Intervals and the Real Line

The real numbers can be visualized as points on a line, called the real number line. Intervals are subsets of the real line defined by inequalities.

• Open Interval: $(a, b) = \{ x \mid a < x < b \}$

• Closed Interval: $[a, b] = \{ x \mid a \leq x \leq b \}$

• Half-Open Intervals: $[a, b)$ or $(a, b]$

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

3. Set Notation

Sets are collections of numbers. Common notations include:

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

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

• $\mathbb{Z}$: The set of all integers

• $A \cap B$: The set of elements in both $A$ and $B$ (intersection)

• $A \cup B$: The set of elements in $A$ or $B$ (union)

4. Functions

A function is a rule that assigns to each element $x$ in a set called the domain exactly one element $y$ in a set called the range. We write $y = f(x)$.

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

• Range: The set of all possible values $f(x)$ can take.

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

5. Graphing a Function

The graph of a function is the set of all points $(x, f(x))$ in the plane. The domain is the set of $x$-values for which the function is defined, and the range is the set of $y$-values the function attains.

• Linear Functions: Functions of the form $f(x) = mx + b$ are called linear functions. Their graphs are straight lines with slope $m$ and $y$-intercept $b$.

• Piecewise Functions: Functions defined by different formulas on different intervals.

Example: The function $f(x) = x^2$ is defined for all real $x$ and its graph is a parabola.

6. Domain and Range: Examples

• For $f(x) = \frac{1}{x}$, the domain is $x \neq 0$ (since division by zero is undefined).

• For $f(x) = \sqrt{x-2}$, the domain is $x \geq 2$.

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 is called the vertical line test.

8. Functions in Real Life

Functions are used to describe relationships in science, engineering, and everyday life. For example, the distance an object travels as a function of time, or the temperature as a function of location.

9. Table: Types of Numbers

The following table summarizes the main types of numbers discussed:

Additional info: These notes are based on the first chapter of a standard college Calculus I course, introducing foundational concepts necessary for further study in calculus, such as limits, derivatives, and integrals.