Numbers and Functions

1. What is a Number?

Calculus is built on the concept of real numbers and functions of real variables. Understanding the types of numbers and their properties is foundational for further study.

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

• Zero: The integer 0.

• Negative Integers: ..., -3, -2, -1

• Rational Numbers: Numbers that can be written as a ratio of two integers, $\frac{m}{n}$, where $n \neq 0$.

• Decimal Expansions: Every rational number has a decimal expansion that either terminates or repeats.

• 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 (possibly infinite) decimal expansions.

Example: $\frac{1}{3} = 0.333...$ (repeating decimal), $\sqrt{2} = 1.4142135...$ (non-repeating, non-terminating decimal)

2. The Real Number Line and Intervals

The real numbers can be visualized as points on a continuous 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 $x$ and $y$ is $|x - y|$.

Example: The interval $[0,1]$ includes all real numbers between 0 and 1, including the endpoints.

3. Set Notation

Sets are collections of numbers or objects. In calculus, we often use set notation to describe intervals and domains.

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

• 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 $B$.

Example: $B = \{ x \mid x^2 - 1 > 0 \}$ is the set of all $x$ such that $x^2 > 1$, i.e., $x < -1$ or $x > 1$.

4. Functions

A function is a rule that assigns to each element $x$ in a set (the domain) exactly one element $y$ (the value of the function at $x$).

• Definition: A function $f$ from a set $A$ to a set $B$ is a rule that assigns to each $x \in A$ a unique $y \in B$, denoted $f(x) = y$.

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

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

• Graph of a Function: The set of points $(x, f(x))$ in the plane.

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

5. Linear Functions

Linear functions are functions of the form $f(x) = mx + b$, where $m$ and $b$ are constants.

• Slope ($m$): The rate of change of the function; rise over run.

• Y-intercept ($b$): The value of $f(x)$ when $x = 0$.

Equation of a line through points $(x_1, y_1)$ and $(x_2, y_2)$:

$\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$ $\displaystyle y - y_1 = m(x - x_1)$

6. Domain and Range

To find the domain of a function, determine all $x$ for which 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 $x \geq 0$.

7. The Vertical Line Test

A graph in the $xy$-plane represents a function if and only if no vertical line intersects the graph more than once. This ensures that each $x$ has only one $y$ value.

8. Functions in Real Life

Functions are used to model relationships in the real world, such as distance over time, population growth, and physical laws.

• Example: The distance $d$ traveled at constant speed $v$ in time $t$ is $d = vt$.

9. Piecewise Functions

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

• Example: The absolute value function:

  • $|x| = x$ if $x \geq 0$

  • $|x| = -x$ if $x < 0$

10. Exercises and Applications

• Find the 200th digit after the period in the decimal expansion of $\frac{1}{7}$.

• Which of the following fractions have finite decimal expansions?

• Draw the following sets of real numbers on the real line.

• Suppose $A$ and $B$ are intervals. Is it always true that $A \cap B$ is an interval? What about $A \cup B$?

• Consider the set $M = \{ x \mid |x| > 0 \}$.

11. Table: Types of Numbers

Additional info: These notes provide the foundational concepts for calculus, including the types of numbers, set notation, functions, and their properties. Understanding these is essential for further topics such as limits, derivatives, and integrals.