Identify Functions - Algebra-2

1. Fundamental Concepts

Function: A special type of relation where each element in the domain corresponds to exactly one element in the range.
- In simple terms, a function requires that each input value (x-value) corresponds to only one output value (y-value), but multiple input values can correspond to the same output value.
Relation vs. Function:
- All functions are relations, but not all relations are functions.
- A relation only needs to satisfy the "set of input-output pairs", while a function adds the restriction of "single-value correspondence" on this basis.

Core of Judgment: Whether each x-value in the domain corresponds to a unique y-value. If a certain x-value corresponds to multiple y-values, the relation is not a function.
Judgment Bases for Different Representations:
- Ordered pairs/tables: Check if there are duplicate x-values corresponding to different y-values.
- Mapping diagrams: Whether each input value (element in the domain) points to only one output value (element in the range).
- Graphs: Judged by the Vertical Line Test (see techniques below).

Example 1: Determine whether the following sets of ordered pairs are functions.
A. {(-2, 3), (0, 5), (2, 7)}
B. {(1, 4), (1, 5), (2, 6)}

Analysis:
- In A, the x-values (-2, 0, 2) are all unique, and each corresponds to one y-value → It is a function.
- In B, x=1 corresponds to two y-values (4 and 5) → It is not a function.

Example 2: Determine whether the following are functions based on the mapping diagrams.

Mapping Diagram 1: Inputs {1, 2, 3} point to output {5, 5, 5} respectively.
Mapping Diagram 2: Inputs {1, 2, 1} point to outputs {3, 4, 5} respectively.
Analysis:
- In Mapping Diagram 1, each input corresponds to only one output (multiple inputs can share the same output) → It is a function.
- In Mapping Diagram 2, input 1 corresponds to two outputs (3 and 5) → It is not a function.

Example 3: Determine whether the following are functions based on the graphs (using the vertical line test).

Graph 1: A straight line y=2x+1
Graph 2: A circle x²+y²=4
Analysis:
- In Graph 1, any vertical line intersects the line at only one point → It is a function.
- In Graph 2, the vertical line x=0 intersects the circle at (0, 2) and (0, -2) → It is not a function.

For Ordered Pairs/Tables:
- List all x-values and check if there are duplicate x-values corresponding to different y-values. If there are duplicates with different y-values, it is not a function.
For Mapping Diagrams:
- Observe whether each input element (domain) points to only one output element (range) through a single arrow. If an input points to multiple outputs, it is not a function.
For Graphs (Vertical Line Test):
- Step 1: Draw several vertical lines (parallel to the y-axis) on the graph.
- Step 2: If all vertical lines intersect the graph at no more than one point → It is a function.
- Step 3: If there exists one vertical line that intersects the graph at two or more points → It is not a function.
Notes:
- Functions allow "many-to-one" (multiple x-values corresponding to the same y-value) but not "one-to-many" (one x-value corresponding to multiple y-values).
- For equations (such as y=ax+b or x=y²), you can judge by transformation: If solving for y results in multiple values corresponding to the same x (e.g., y=±√x in x=y²), then it is not a function.