Numbers and Functions

What is a Number?

Calculus begins with understanding the types of numbers used in mathematics, especially real numbers. These form the basis for functions and further concepts in calculus.

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

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

• Zero: $0$

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

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

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

Decimal Expansions: Rational numbers have either terminating or repeating decimals. Irrational numbers have non-repeating, non-terminating decimals.

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

The Real Number Line and Intervals

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

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

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

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

Set Notation

Sets are collections of numbers. Interval notation and set-builder notation are used to describe sets of real numbers.

• Example: $\{x \mid 1 < x < 6\}$ 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 $B$.

Functions

Definition of a Function

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. The function is often written as $f(x)$.

• Domain: The set of all possible input values ($x$) for which the function is defined.

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

• Example: $f(x) = x^2$ has domain $(-\infty, \infty)$ and range $[0, \infty)$.

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 $y = x^2$ passes the vertical line test, but the graph of a circle $x^2 + y^2 = r^2$ does not.

Linear Functions

A linear function is given by the formula:

• $f(x) = mx + b$

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

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

• Example: $f(x) = 2x + 3$

Domain and Range from Formulas

To find the domain and range of a function given by a formula, determine for which $x$ the formula makes sense (e.g., no division by zero, no square roots of negative numbers).

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

• Example: $f(x) = \sqrt{x}$ has domain $x \geq 0$.

Functions in Real Life

Functions model relationships in science, engineering, and everyday life. For example, the distance $d$ from a point to the origin is a function of its coordinates:

• $d = \sqrt{x^2 + y^2}$

Piecewise Defined Functions

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

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

Tables

Comparison of Number Types

Interval Notation Examples

Additional info:

• This summary covers the foundational material for a first-semester calculus course, including numbers, functions, graphs, and set notation. These concepts are essential for understanding limits, derivatives, and integrals in later chapters.