How do you know if a relation is a function?

The most important property is that it should satisfy the vertical line test.

This means that if it is possible to draw a vertical line through some point on the graph of the relation such that it intersects the graph at more than one point, it is NOT a function.

Mathematically, it means that every element in the domain is mapped by the relation to one single value in the range.

Example of relations that are not functions:
y=&sqrt(x)
y=arcsin(x)

Examples of functions:
y=x
y=x²
all polynomials
y=sin(x)
y=ex

To determine if a relation is a function, you need to examine the relationship between the input values (domain) and the output values (range) of the relation. Here's a step-by-step approach to check if a relation is a function:

1. Understand the concept: A function is a special type of relation where each input (domain value) is associated with only one output (range value). In other words, for a relation to be a function, each input cannot have multiple outputs.

2. Identify the input and output values: Look at each pair of values within the relation. Usually, the input values are given in the domain of the relation, and the output values are given in the range.

3. Check for multiple outputs: Examine each input value in the domain column and see if it appears more than once. If an input value is associated with different output values, the relation is not a function.

4. Use a visual representation: Create a scatter plot or graph to represent the relation. If all the points lie on a vertical line, with no two points having the same x-coordinate, then the relation is a function. On the other hand, if there is a vertical line crossing the graph at more than one point with the same x-coordinate, the relation is not a function.

5. Use the vertical line test: Imagine a vertical line drawn anywhere on the graph representing the relation. If this vertical line intersects the graph at more than one point, the relation is not a function. However, if the line intersects the graph at only one point for every possible vertical line, the relation is a function.

By following these steps, you can determine whether a given relation is a function or not.