Functions and Relations in Algebra

Study Notes

Functions

A function is a special type of relation where each input has exactly one output.
Functions can be represented in various ways:
- Verbally (describing the relationship in words)
- Numerically (using a table of values)
- Graphically (using a coordinate plane)
- Algebraically (using an equation)
Independent variable: The input value (often x).
Dependent variable: The output value (often y).
Vertical Line Test: A graph represents a function if no vertical line intersects the graph more than once.

Relations

A relation is any set of ordered pairs.
Ordered pairs are represented as (input, output).
Relations can be expressed in various ways like functions, but don't require each input to have only one output.

Key Differences between Functions and Relations

Functions: Each input value maps to exactly one output value.
Relations: Input values can map to one or more output values. (A relation that is not a function).

Function Notation

Function notation uses a letter, such as f, to represent the function name paired with input variable like x to indicate the output of a function.
e.g., f(x) = 2x + 1
This notation shows that f is the name of the function and the value substituted for x gives the output.

Identifying Functions from Representations

Tables: Look for inputs to be unique; if an input value appears more than once, with a different output value, it is not a function.
Graphs: Apply the vertical line test.
Equations: Check if each input value leads to one and only one output value. For example,
y = 2x + 5 is a function; each x creates only one y;
x^2 + y^2 = 9 is not usually a function (it is a relation) since a given x value may have two y outputs.

Domain and Range

Domain: The set of all possible input values (x-values) for a function or relation.
Range: The set of all possible output values (y-values) for a function or relation.

Examples of Functions and Relations

Function Example:
Input values {1, 2, 3} produce output values {3, 4, 5} -> A function, since each input produces exactly one output
Relation Example:
Input values {1, 2, 2} produce output values {3, 4, 5} -> A relation that is not a function-- the input '2' has two different outputs.

Identifying Functions from Word Problems

Some word problems describe situations that can be represented by functions.
Use the key features of functions—a single input resulting in a single output—to determine if the problem describes a function. e.g.: Cost of pizza is a function of the number of toppings, each topping will add a fixed amount; every number of toppings will have exactly one associated price. Note, if a customer could choose any number of toppings, and any combination of toppings, that might or might not be a function. If it is a function, all combinations of toppings, and numbers of toppings, will be unique costs.

Description

This quiz explores the concepts of functions and relations in algebra. It covers definitions, characteristics, and methods of representation, including verbal, numerical, graphical, and algebraic forms. Test your understanding of key differences between functions and relations, and the criteria for identifying functions.