Numbers and Functions

1. What is a Number?

Calculus relies on a solid understanding of different types of numbers and their properties. This section introduces the main classes of numbers used in calculus and their significance.

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

• Zero: The number 0, which is neither positive nor negative.

• Negative Integers: ..., -3, -2, -1

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

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

• Real Numbers: The set of all rational and irrational numbers. Every real number can be represented by a (possibly infinite) decimal expansion.

Example: $\frac{1}{3} = 0.3333\ldots$, $\sqrt{2} = 1.4142135\ldots$

1.1 Why are Real Numbers Called 'Real'?

• The term 'real' distinguishes these numbers from 'imaginary' numbers (involving $\sqrt{-1}$).

• Real numbers can be visualized as points on a continuous number line.

1.2 The Real Line and Intervals

• The real line is a geometric representation of all real numbers as points on a straight line.

• Intervals are subsets of the real line, such as $[a, b]$ (all $x$ with $a \leq x \leq b$), $(a, b)$ (all $x$ with $a < x < b$), etc.

• 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.

1.3 Set Notation

• Sets are collections of numbers, often described using set-builder notation, e.g., $\{x \mid a < x < b\}$.

• Common sets include $\mathbb{R}$ (all real numbers), $\mathbb{Q}$ (all rational numbers), $\mathbb{Z}$ (all integers).

Example: $A = \{x \mid 1 < x < 6\}$ is the set of all real numbers between 1 and 6.

2. Functions

Functions are fundamental objects in calculus, describing how one quantity depends on another.

2.1 Definition of a Function

• A function $f$ assigns to each input $x$ in its domain a unique output $f(x)$.

• The domain of a function is the set of all $x$ for which $f(x)$ is defined.

• The range is the set of all possible values $f(x)$ can take.

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

2.2 Graphing a Function

• The graph of a function is the set of all points $(x, f(x))$ in the plane.

• To graph a function, plot points for various $x$ and connect them smoothly if the function is continuous.

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

2.3 Linear Functions

• A linear function has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• The graph is a straight line.

Example: $f(x) = 2x + 1$ is a line with slope 2 and y-intercept 1.

2.4 Domain and Range from Formulas

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

• The range is found by considering all possible outputs for $x$ in the domain.

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

2.5 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.

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

2.6 Functions in Real Life

• Functions model relationships between quantities, such as distance over time, population growth, or temperature changes.

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

3. Summary Table: Types of Numbers

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