Mathematics Relations and Functions

Relations

• A relation is a set of ordered pairs.
• Ordered pairs are written as (input, output) where the input comes from a set called the domain, and the output comes from a set called the range.
• The domain is the set of all possible input values.
• The range is the set of all possible output values.
• Relations can be represented in different ways:
  • Using a set of ordered pairs.
  • Using a mapping diagram.
  • Using a graph.
  • Using a table.
• Example: {(1, 2), (2, 4), (3, 6)} is a relation where the domain is {1, 2, 3} and the range is {2, 4, 6}.

Functions

• A function is a special type of relation where each input value (from the domain) corresponds to exactly one output value (in the range).
• In other words, no two different ordered pairs can have the same first element.
• Functions can be represented in the same ways as relations.
• Vertical Line Test: A graph represents a function if any vertical line intersects the graph at most once.

Mapping Diagrams

• Mapping diagrams visually represent relations and functions.
• Arrows connect elements in the domain to elements in the range.
• For a relation to be a function, each element in the domain must have exactly one arrow pointing to an element in the range.

Function Notation

• Function notation is used to represent functions symbolically.
• Common notation: f(x), where 'f' represents the function and 'x' represents the input value.
• f(x) represents the output value of the function for the input value 'x'.

Types of Functions

• Linear Functions: have a constant rate of change and can be represented by a straight line.
• Quadratic Functions: have a squared term and can be represented by a parabola.
• Cubic Functions: contain a cubed term.
• Polynomials: generalize functions made up of sums and powers of x.
• Exponential Functions: contain an exponent where x is the exponent.

Domain and Range of Functions

• Domain: The set of all possible input values for which the function is defined.
• Range: The set of all possible output values that the function can produce.

Identifying Functions

• When given a table, consider if every input (x-value) produces only one output (y-value).
• When examining a set of ordered pairs, ensure each x-value is associated with only one y-value.

Determining if a Relation is a Function

• Use the vertical line test for graphs.
• Check if each x-value in a set of ordered pairs or a table is paired with only one y-value.
• Examine the mapping diagram; each element in the domain is assigned to only one element of the range.

Key Differences Between Relations and Functions

• Relations allow multiple outputs (y-values) to correspond to a single input (x-value).
• Functions ensure unique and single outputs(y-values) for each input (x-value).

Examples

• Consider the relation {(1, 2), (2, 3), (1, 4)}. This is not a function because the input '1' is associated with two different outputs (2 and 4).
• The relation {(1, 2), (2, 3), (3, 4)} is a function because each input is mapped to a unique output.

Applications of Functions and Relations

• Real-world situations can be modeled by functions and relations.
• For example, the relationship between distance traveled and time can often be described by a linear function.