Relations and functions are two types of mathematical concepts that are closely related but have distinct characteristics.

Types of Relations

Relations are any set of (x, y) pairs that describe the relationship between two quantities. Every function is a relation, but not every relation is a function. They can be represented in various ways, such as through set mapping (using numbers and arrows), lists of ordered pairs, graphs, tables, words, and equations.

Types of Functions and Equations

Functions are a type of relation where for each input, there exists exactly one output. It is a rule that describes the relationship between two quantities. Unlike relations, functions have a specific set of input and output values that follow a certain rule or pattern. Functions can also be represented through various methods such as set mapping, lists of ordered pairs, graphs, tables, words, and equations.

Domain and Range of a Function

The domain of a function is the set of all input values (x-values), while the range of a function is the set of all output values (y-values). To express inequality, interval notation is often used, with parentheses ( or ) for not included, and square brackets [ or ] for included values. Vertical and horizontal intervals indicate different directions and movements of the function on a graph.

Transformations of Parent Functions

Transformations of parent functions include translations, reflections, and dilations. These transformations alter the graph of the function in different ways. Translations shift the graph vertically or horizontally, reflections flip the graph across a line, and dilations change the shape of the graph through vertical stretch or compression.

Relations vs Functions Examples

A common way to differentiate between relations and functions is by testing if a set of points or a graph represents a function. If x-values are repeated in the set of points, the relation is not a function. Similarly, from a graph, drawing a vertical line and checking if it intersects the graph at more than one point indicates that the relation is not a function.

In conclusion, understanding the differences between relations and functions, as well as their various representations and transformations, is essential in the study of mathematics and applied sciences. With the use of appropriate examples, exercises, and practice, students can improve their comprehension and proficiency in these fundamental concepts.