Relations and Functions

Flashcards covering the vocabulary and core concepts of relations, functions, coordinate systems, and function notation based on the lecture notes.

Relation

A set of ordered pairs.

Ordered pair

A pair of numbers used to name a point in the coordinate plane, written in the form $(x, y)$, where each point is named by exactly one pair.

Domain

The set of all first coordinates ($x$-coordinates) from the ordered pairs in a relation.

Range

The set of all second coordinates ($y$-coordinates) from the ordered pairs in a relation.

Cartesian coordinate system

A system made up of a horizontal $x$-axis and a vertical $y$-axis that meet at the origin $(0, 0)$, dividing the plane into four quadrants.

Origin

The point $(0, 0)$ where the $x$-axis and $y$-axis meet.

Quadrant

One of the four areas of the Cartesian coordinate plane; the sign of the coordinates ($(+,+)$, $(-,+)$, $(-,-)$, or $(+,-)$) determines which area a point is in.

Mapping Diagram

A representation of a relation that uses ovals and arrows to show how each element of the domain is paired with elements in the range.

Four Ways to Represent Relations

The four methods are: Ordered Pairs, Mapping Diagram, Graph, and Table of Values.

Function

A special type of relation in which each element of the domain is paired with exactly one element in the range; no two ordered pairs have equal abscissas.

Abscissa

The first coordinate ($x$-value) in an ordered pair.

One-to-One Correspondence

A mapping where each element of the domain maps to exactly one unique element in the range.

Many-to-One Correspondence

A mapping where multiple elements from the domain map to the same element in the range; this is still considered a function.

One-to-Many Correspondence

A mapping where one element of the domain maps to more than one element in the range; this is not a function.

Vertical Line Test

A test used to determine if a graph represents a function; if any vertical line intersects the graph in more than one point, the relation is not a function.

Independent variable

The variable (usually $x$) whose values make up the domain of a function.

Dependent variable

The variable (usually $y$) whose values make up the range of a function, as its values depend on the input of the independent variable.

Function Notation

A notation used to designate a function, replacing "$y$" with $f(x)$, which is pronounced "$f$ of $x$".