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 study in calculus.

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

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

• Zero: $0$

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

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

• Real Numbers: All rational and irrational numbers, including decimals and numbers with infinite non-repeating decimal expansions.

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

The Real Number Line and Intervals

The real number line is a visual representation of all real numbers. Intervals are subsets of the real number line, often used to specify domains and ranges of functions.

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

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

• Half-Open Interval: $[a, b)$ or $(a, b]$ includes one endpoint.

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

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

Set Notation

Sets are collections of numbers or objects. In calculus, sets are used to define domains, ranges, and solution sets.

• Set of all $x$ such that $a \leq x \leq b$: $\{x \mid a \leq x \leq b\}$

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

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

Example: $A = \{x \mid x > 1\}$, $B = \{x \mid x < 3\}$, $A \cap B = \{x \mid 1 < x < 3\}$

Functions

Definition of a Function

A function is a rule that assigns to each input value $x$ in its domain exactly one output value $f(x)$.

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

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

• Notation: $f: D \to R$, where $D$ is the domain and $R$ is the range.

Example: $f(x) = x^2$ has domain $\mathbb{R}$ 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.

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

Example: The graph of $f(x) = x^2$ is a parabola.

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 are used to model relationships in science, engineering, and everyday life. For example, the distance an object travels over time can be described by a function.

• Distance Function: $d(t)$ gives the distance at time $t$.

• Temperature Function: $T(x)$ gives the temperature at location $x$.

Piecewise Defined Functions

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

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

Summary Table: Types of Numbers

Summary Table: Types of Intervals

Additional info:

• Piecewise functions and the vertical line test are foundational for understanding function graphs and their properties.

• Real numbers are called 'real' because they can be represented on the number line, unlike imaginary numbers.