Numbers and Functions

1. What is a Number?

This section introduces the foundational concept of numbers, which are the building blocks of calculus. Understanding different types of numbers is essential for working with functions and calculus concepts.

• Positive Integers: The simplest numbers, such as 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, e.g., $\frac{1}{2}$, $-\frac{3}{4}$, $5$ (since $5 = \frac{5}{1}$).

• 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 point on the number line corresponds to a real number.

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

Example:

• $\frac{1}{3} = 0.3333\ldots$ (repeating decimal)

• $\sqrt{2} = 1.4142135\ldots$ (non-repeating, non-terminating decimal)

2. 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 and are used to describe domains and ranges of functions.

• Closed Interval: $[a, b]$ includes all numbers $x$ such that $a \leq x \leq b$.

• Open Interval: $(a, b)$ includes all numbers $x$ such that $a < x < b$.

• Half-Open Intervals: $[a, b)$ or $(a, b]$ include one endpoint but not the other.

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

Example:

• The interval $(-1, 2]$ contains all $x$ such that $-1 < x \leq 2$.

3. Set Notation

Set notation is used to describe collections of numbers, such as intervals or solution sets.

• 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 all elements in $A$ or $B$ (or both).

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

Example:

• $A = \{x \mid 1 < x < 3\}$, $B = \{x \mid 2 < x < 4\}$, then $A \cap B = \{x \mid 2 < x < 3\}$.

4. Functions

A function is a rule that assigns to each input (from a set called the domain) exactly one output (from a set called the range). Functions are central objects in calculus.

• Definition: To specify a function $f$, you must:

  • Give a rule for how to compute $f(x)$ for each $x$ in the domain.

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

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

• Range: The set of all possible output values.

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

Example:

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

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

5. 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 $f$. Graphs help visualize the behavior of functions.

• Linear Functions: $f(x) = mx + b$ is a straight line with slope $m$ and $y$-intercept $b$.

• 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 $f(x) = x^2$ is a parabola opening upwards.

• The graph of $f(x) = \sqrt{x}$ is defined only for $x \geq 0$.

6. Domain and Range: Examples

Finding the domain and range of a function is a key skill in calculus. The domain is determined by the set of $x$-values 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 $x \neq 0$ (since division by zero is undefined).

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

7. Functions in Real Life

Functions are used to describe relationships in science, engineering, economics, and everyday life. For example, the distance an object travels as a function of time, or the cost of goods as a function of quantity purchased.

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

8. Summary Table: Types of Numbers

The following table summarizes the main types of numbers discussed:

Additional info: The table is inferred from the text and standard mathematical classification.

Key Formulas and Properties

• Distance on the real line: $|a - b|$

• Linear function: $f(x) = mx + b$

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

Conclusion

This chapter lays the groundwork for calculus by introducing numbers, the real number line, set notation, and the concept of functions. Mastery of these ideas is essential for understanding limits, derivatives, and integrals in subsequent chapters.