Question

how can i tell if plotted points are functions and what is a commute

Answer 1

To determine if a set of plotted points represents a function, you can use the **vertical line test**. Here's how it works:

1. **Vertical Line Test**: If you can draw a vertical line at any position on the x-axis that intersects the plotted points in more than one place, then the plotted points do not represent a function. This is because, for a relation to be a function, each input (x-value) must correspond to exactly one output (y-value).

2. **For example**: Consider the following sets of points:
- **Function**: (1, 2), (2, 3), (3, 5) – A vertical line drawn anywhere will only intersect at one point.
- **Not a function**: (1, 2), (1, 3), (2, 4) – A vertical line drawn at \(x = 1\) will intersect the points at both (1, 2) and (1, 3), so this does not represent a function.

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Now, regarding your second question about what a **commute** means:

In mathematics, particularly in the context of operations or functions, "commute" refers to the property of being able to change the order of operations without affecting the result. Specifically, if you have two operations (or functions) \(f\) and \(g\):

- They **commute** if:
\[ f(g(x)) = g(f(x)) \]
for all \(x\) in the domain.

A common example is the addition of numbers (e.g., \(a + b = b + a\)) and multiplication (e.g., \(a \cdot b = b \cdot a\)), which are both commutative operations. However, not all operations commute; for example, subtraction and division do not meet this property.

If you meant "commute" in a different context, please feel free to clarify!