Function Identification and Characteristics
Identification of Functions
Definition of Relation
A relation is defined as any set of ordered pairs.
Definition of Function
A function is a special type of relation that fulfills the criteria of mapping each domain element (input, denoted as X) to exactly one range element (output, denoted as Y).
Key Characteristics:
Each input (X value) has one unique output (Y value).
Notation: If a function is represented as f, then for each input x, there exists a unique output f(x).
Examples of Functions
Example of Function (Set A)
The mapping of values is as follows:
When X = 1, Y = 2
When X = 2, Y = 5
When X = 3, Y = 8
When X = 4, Y = 11
Identification as a Function:
Since each input (X) corresponds to one unique output (Y), this is a function.
Example of Function (Set B)
The mapping of values is:
When X = 7, Y = 3
When X = 4, Y = 0
When X = 8, Y = 1
When X = -8, Y = 7
Identification as a Function:
Each input is paired with exactly one output, confirming it as a function.
Example of Non-Function (Set C)
Issues arise with the mapping when X = 0:
Possible outputs are Y = 1 or Y = 2.
Identification as Non-Function:
The presence of two outputs for a single input (X = 0) indicates that it does not satisfy the uniqueness criterion of a function.
Example of Function (Set D)
The mapping includes:
When X = -2, Y = 10
When X = -1, Y = 7
When X = 0, Y = 10
When X = 1, Y = 13
Identification as a Function:
Each input value maps to one unique output, thereby confirming it as a function.
Relations as Ordered Pairs
Ordered pairs are assessed for uniqueness of output for each input.
Evaluation of Set A
Ordered pairs provided: {(2, 3), (8, 4), (2, 5)}.
Analysis:
The input X = 2 maps to Y values of 3 and 5.
Conclusion:
This is not a function due to the lack of uniqueness in the output for the input 2.
Evaluation of Other Sets
Set B: Ordered pairs demonstrate that each X only maps to a single Y, confirming it as a function.
Set C and Set D: Each also demonstrates uniqueness for their respective X values in its relationships to Y outputs.
Mapping Diagrams
A mapping diagram visually represents the relationships between inputs (X) and outputs (Y).
Characteristics:
Inputs are placed on the left, outputs on the right.
Each input is connected to its corresponding output.
Example (Mapping Diagram Set A)
X values: 1, 2, 3; Y values: -2, 3, 8.
Identification as a Function:
Each input maps to one and only one output.
Example (Mapping Diagram Not a Function)
X = 0 connecting to Y values -1 and 8.
Conclusion:
X has more than one output, thus it is not a function.
Real-world Analogy with Pizza Delivery
Domain: Pizza delivery places (e.g., Pizza Hut, Domino's, Papa John's, Little Caesars).
Possible scenario:
Each pizza delivery person delivering to different houses example.
Each delivery person can only deliver to one house at a time, which follows the function definition (unique outputs).
Conflicting Scenario
Situation where Pizza Hut and Domino's both deliver to the same house.
Still a function because they are delivering to the same address but doing so individually.
Unique Questions
Could one person deliver to two places?
No, invalidating function properties as it violates the unique mapping.
Vertical Line Test
A test used to determine if a graph is a function.
Draw a vertical line through the graph:
If the line intersects the graph at exactly one point at all times, the graph represents a function.
If the line intersects at more than one point at any place on the graph, it is not a function.
Applications of the Vertical Line Test
Examples A, B, C satisfy the function criteria using the vertical line test.
Example D fails the test, as it intersects the graph at multiple points for the same X value, indicating multiple outputs.
Summary
For a relation to qualify as a function, each input must correspond to one and only one output.
The vertical line test serves as a reliable graphical method to assess functions in visual representation. Understanding these principles is essential for identifying and validating functions in various mathematical contexts.