Pre-Calculus: Real Numbers, Cartesian Plane, and Functions

Vocabulary flashcards covering the Real Number System, Cartesian Plane, and Functions based on lecture transcript notes.

Real Number System

A set that contains elements called real numbers.

Set

A well-defined collection of objects.

Element

A member or individual object belonging to a set.

Real Number

Any number that can be plotted on the number line.

Natural Numbers

Defined as the counting numbers, denoted by $\mathbb{N}$ and listed as $\mathbb{N} = \{1, 2, 3, 4, 5, 6, 7, 8, \dots\}$.

Whole Numbers

Defined as the counting numbers together with zero, denoted by $\mathbb{W}$ and listed as $\mathbb{W} = \{0\} \cup \mathbb{N} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, \dots\}$.

Integers

Defined as the natural numbers together with their negatives and zero, denoted by $\mathbb{Z}$ and listed as $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

Rational Numbers

Defined as numbers that can be expressed as a ratio of two integers, denoted by $\mathbb{Q}$ and listed as $\mathbb{Q} = \left\{\frac{a}{b} \;\middle|\; a, b \in \mathbb{Z}, b \neq 0\right\}$.

Symbol $\in$

A mathematical symbol read as 'an element of'.

Irrational Numbers

Defined as numbers that cannot be expressed as fractions, denoted by $\mathbb{Q}'$ and listed as $\mathbb{Q}' = \{x \mid x \notin \mathbb{Q}\}$.

Real Numbers Set Definition

Defined as the union of rational and irrational numbers, denoted by $\mathbb{R}$ and listed as $\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'$.

Ellipsis

A series of three dots ($\dots$) used in set notation to indicate that a pattern continues indefinitely.

Non-integer Rational Numbers

Rational numbers that cannot be simplified to integers, commonly referred to as fractions.

Negative Integers

The set of integers less than zero, represented as $\{\dots, -3, -2, -1\}$.

Real Number Hierarchy Diagram

A structural diagram demonstrating how Natural Numbers, Zero, Negative Integers, Fractions, Rational Numbers, and Irrational Numbers form Real Numbers.

Addition of Real Numbers

A primary binary operation in the real number system denoted by $+$, yielding the sum $p + q$.

Multiplication of Real Numbers

A primary binary operation in the real number system denoted by $\cdot$, yielding the product $p \cdot q$.

Subtraction of Real Numbers

An operation defined using addition where $p - q = r$ if and only if $p = q + r$.

Division of Real Numbers

An operation defined using multiplication where $p \div q = r$ if and only if $p = q \cdot r$, given that $q \neq 0$.

Division by Zero

An operation that is not defined because no real number $r$ can satisfy $p = 0 \cdot r$ when $p \neq 0$.

Indeterminate Expression $0 \div 0$

An expression resulting from division of zero by zero, which is indeterminate because $0 = 0 \cdot r$ holds for any real number $r$.

Product of Two Real Numbers Rule

The product of two real numbers is 'Sometimes' positive or negative, because the product can equal zero.

Sum of Two Integers Rule

The sum of two integers is 'Always' an integer.

Difference of Two Whole Numbers Rule

The difference of two whole numbers is 'Sometimes' a whole number, failing when the result is negative.

Quotient of Two Integers Rule

The quotient of two integers is 'Sometimes' an integer, as division may yield a non-integer fraction.

Sum of Two Irrational Numbers Rule

The sum of two irrational numbers is 'Sometimes' an irrational number (for example, $\sqrt{2} + (-\sqrt{2}) = 0$, which is rational).

Product of Two Irrational Numbers Rule

The product of two irrational numbers is 'Sometimes' irrational (for example, $\sqrt{2} \cdot \sqrt{2} = 2$, which is rational).

Rational and Irrational Intersection Rule

A real number is 'Never' both rational and irrational because $\mathbb{Q}$ and $\mathbb{Q}'$ share no common elements.

Decimal Expansion of Rational Numbers Rule

The decimal expansion of a rational number 'Sometimes' terminates, as it may either terminate or repeat.

Square Root of Real Numbers Rule

Every real number 'Sometimes' has a square root in the real number system, as negative real numbers do not have real square roots.

Expressing Numbers as Fractions Rule

Every number 'Sometimes' can be expressed as a fraction, which holds for rational numbers but fails for irrational numbers.

Commutative Property of Addition

States that changing the order of addends does not affect the sum: $a + b = b + a$.

Commutative Property of Multiplication

States that changing the order of factors does not affect the product: $a \cdot b = b \cdot a$.

Commutative Property

The principle that the order in which numbers are taken does not affect the sum or product.

Associative Property of Addition

States that changing the grouping of addends does not affect the sum: $(a + b) + c = a + (b + c)$.

Associative Property of Multiplication

States that changing the grouping of factors does not affect the product: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.

Associative Property

The principle that the grouping in which numbers are taken does not affect the sum or product.

Additive Identity Element

The element $0$, which satisfies $a + 0 = a$ for any real number $a$.

Multiplicative Identity Element

The element $1$, which satisfies $a \cdot 1 = a$ for any real number $a$.

Identity Property of Addition

Performing addition between any number $a$ and the identity element $0$ results in $a$.

Identity Property of Multiplication

Performing multiplication between any number $a$ and the identity element $1$ results in $a$.

Additive Inverse

The negative of $a$, denoted $-a$, which satisfies $a + (-a) = 0$.

Multiplicative Inverse

The reciprocal of a non-zero number $a$, denoted $\frac{1}{a}$, which satisfies $a \cdot \left(\frac{1}{a}\right) = 1$.

Left-hand Distributive Property

States that for any real numbers $a$, $b$, and $c$: $a \cdot (b + c) = a \cdot b + a \cdot c$.

Right-hand Distributive Property

States that for any real numbers $a$, $b$, and $c$: $(a + b) \cdot c = a \cdot c + b \cdot c$.

Origin (Coordinate Line)

The reference point $O$ on a line assigned the coordinate $0$.

Unit of Length

A standard segment $OP$ used to establish distance along a coordinate line.

Coordinate Line

A line on which a scale is established by assigning real numbers to points.

Coordinate of a Point

The real number representing a specific point on a coordinate line.

Graph of a Number

The point on a coordinate line that corresponds to a given real number.

One-Dimensional Space Representation

A space in which points on a line are represented by single numbers.

Two-Dimensional Space Representation

A space in which points in a plane are represented by ordered pairs of numbers $(a, b)$.

Cartesian Plane

A two-dimensional coordinate system formed by two perpendicular axis lines.

Perpendicular Axes Convention

The rule in a coordinate plane stating that the horizontal and vertical axes meet at right angles.

X-axis

The horizontal coordinate line with positive coordinates to the right of the origin.

Y-axis

The vertical coordinate line with positive coordinates above the origin.

Quadrants

The four regions into which the coordinate axes separate the plane.

Quadrant I

The region of the Cartesian plane where both $x$ and $y$ coordinates are positive.

Quadrant II

The region of the Cartesian plane where $x$ is negative and $y$ is positive.

Quadrant III

The region of the Cartesian plane where both $x$ and $y$ coordinates are negative.

Quadrant IV

The region of the Cartesian plane where $x$ is positive and $y$ is negative.

Point $A(3,1)$ Location

A point located in Quadrant I of the Cartesian plane.

Point $B(-1,2)$ Location

A point located in Quadrant II of the Cartesian plane.

Point $C(-2,-3)$ Location

A point located in Quadrant III of the Cartesian plane.

Point $D(2,-3)$ Location

A point located in Quadrant IV of the Cartesian plane.

Point $E(0,3)$ Location

A point located on the positive y-axis.

Point $F(-2,0)$ Location

A point located on the negative x-axis.

Run ($\Delta x$)

The horizontal increment between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, defined as $\Delta x = x_2 - x_1$.

Rise ($\Delta y$)

The vertical increment between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, defined as $\Delta y = y_2 - y_1$.

Slope ($m$)

The ratio of rise to run for a nonvertical line passing through $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, defined as $m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$.

Positive Slope Meaning

Indicates that a line rises uphill to the right.

Negative Slope Meaning

Indicates that a line falls downhill to the right.

Distance Formula in the Plane

A formula derived from the Pythagorean Theorem giving the distance $d$ between $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ as $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Distance between $D(1,4)$ and $E(3,-3)$

The distance value $d = \sqrt{(3-1)^2 + (-3-4)^2} = \sqrt{4 + 49} = \sqrt{53}$.

Slope between $D(1,4)$ and $E(3,-3)$

The slope value $m = \frac{-3 - 4}{3 - 1} = -\frac{7}{2}$.

Distance between $F(-2,0)$ and $G(-1,-3)$

The distance value $d = \sqrt{(-1 - (-2))^2 + (-3 - 0)^2} = \sqrt{1 + 9} = \sqrt{10}$.

Slope between $F(-2,0)$ and $G(-1,-3)$

The slope value $m = \frac{-3 - 0}{-1 - (-2)} = -3$.

Distance between $H(3,1)$ and $I(2,-1)$

The distance value $d = \sqrt{(2 - 3)^2 + (-1 - 1)^2} = \sqrt{1 + 4} = \sqrt{5}$.

Slope between $H(3,1)$ and $I(2,-1)$

The slope value $m = \frac{-1 - 1}{2 - 3} = 2$.

Function (Set Mapping Definition)

A collection of assignments where to each element of a set $A$, a unique element of a set $B$ is assigned.

Domain of a Function

The set $A$ of starting elements in a function mapping $f: A \rightarrow B$.

Range of a Function

The set $B$ (or subset of assigned outputs) containing assigned elements in a function mapping $f: A \rightarrow B$.

Notation $f: A \rightarrow B$

Denotes 'f is a function from A into B', 'f takes A into B', or 'f maps A into B'.

Image of $a$ under $f$

The element $f(a)$ in set $B$ assigned to $a \in A$ by function $f: A \rightarrow B$.

Square Function Mapping $f_1$

The relation $f_1 = \{(-1,1), (0,0), (1,1), (2,4)\}$, which is a valid function mapping domain elements to their squares.

Square Root Relation Mapping $f_2$

The relation $$f_2 = {(0,0), (1,1), (1,-1), (4,2), (4,-2), (9,3), (9,-3)}$, which is not a function because single inputs map to multiple outputs.

Function (Ordered Pairs Definition)

A set of ordered pairs with the property that no two ordered pairs have the same first component and different second components.

Analysis of $f_a = \{(-2,1), (-1,2), (0,0), (-1,2), (-2,2)\}$

Not a function because the first component $-2$ is paired with two different second components ($1$ and $2$).

Analysis of $f_b = \{(-2,1), (-1,2), (0,0), (1,2), (2,1)\}$

A valid function because no two ordered pairs share the same first component with different second components.

Domain Default Rule

If a function is defined by an equation and the domain is not indicated, the domain is assumed to be the set of all real number replacements of the independent variable.

Independent Variable

The variable in a function equation whose replacements form the domain.

Dependent Variable

The variable in a function equation whose values form the range corresponding to domain values.

Domain of $y = x - 3$

The set of all real numbers $\mathbb{R}$.

Domain of $y = x + 5$

The set of all real numbers $\mathbb{R}$.

Domain of $y = \frac{1}{x - 3}$

The set of all real numbers except $x = 3$, written as $\{x \in \mathbb{R} \mid x \neq 3\}$.

Domain of $y = \frac{1}{x + 5}$

The set of all real numbers except $x = -5$, written as $\{x \in \mathbb{R} \mid x \neq -5\}$.

Symbol $f(x)$

Read as 'f of x', representing the number in the range of function $f$ to which domain value $x$ is paired.

Evaluation $f(3)$ for $f(x) = 2x + 1$

The calculated range value $f(3) = 2(3) + 1 = 7$.

Difference Quotient Formula

The expression $\frac{f(x + h) - f(x)}{h}$ where $h \neq 0$.

Composite Function Definition

A new function $(g \circ f): A \rightarrow C$ defined by $(g \circ f)(x) \equiv g(f(x))$, given $f: A \rightarrow B$ and $g: B \rightarrow C$.