๐ What is a Function Table?
A function table represents a mathematical function, where each input value (usually denoted as $x$) has a unique output value (usually denoted as $y$). In other words, for every $x$ you plug in, you get only one possible $y$.
- ๐ข Each input ($x$) is associated with exactly one output ($y$).
- ๐ Graphically, it passes the vertical line test (a vertical line drawn anywhere on the graph will only intersect the graph at one point).
- ๐ It can be represented by an equation, such as $y = f(x)$.
๐ What is a Non-Function Table?
A non-function table is a table where at least one input value ($x$) is associated with multiple output values ($y$). This means for the same $x$, you can get different $y$ values.
- ๐งฎ At least one input ($x$) is associated with more than one output ($y$).
- ๐ Graphically, it fails the vertical line test (a vertical line can intersect the graph at more than one point).
- ๐ซ It cannot be represented by a single function $y = f(x)$ across its entire domain.
๐ Function Table vs. Non-Function Table: The Key Differences
| Feature | Function Table | Non-Function Table |
|---|
| Definition | Represents a function where each input has a unique output. | Represents a relation where at least one input has multiple outputs. |
| Input-Output Relation | One-to-one or many-to-one. | One-to-many. |
| Vertical Line Test | Passes the vertical line test. | Fails the vertical line test. |
| Mathematical Representation | Can be represented by a function $y = f(x)$. | Cannot be fully represented by a single function. |
| Example | $y = x^2$ | $x = y^2$ |
๐ก Key Takeaways
- ๐ง A function table ensures that each input has one and only one output.
- ๐งญ A non-function table allows an input to have multiple outputs.
- ๐งช The vertical line test is a quick visual way to determine if a table (or its graph) represents a function.