In the image you provided, two relations (Relation 3 and Relation 4) are displayed, and you are asked to determine if each relation is a function or not.

To recap the definition of a function:

A relation is a function if every input (domain) is mapped to exactly one output (range). This means that each domain value must correspond to only one unique range value. If any domain value corresponds to more than one range value, the relation is not a function.

Let's analyze both relations:

1. Relation 3 (left side):

   • Domain: {b, g, t, v}
   • Range: {moon, pen}
   • The domain value "g" maps to "pen", "t" maps to "pen", and "v" maps to "pen". Since multiple domain values map to the same range, that’s fine, but each domain value maps to only one range value.
   • Conclusion: Relation 3 is a function.

2. Relation 4 (right side):

   • Domain: {8, -4, 9, -8, 2}
   • Range: {6, -9, 9, 5}
   • All domain values map to exactly one range value. No domain value is repeated with different range values.
   • Conclusion: Relation 4 is also a function.

Final Answer:

• Relation 3: Function
• Relation 4: Function

Do you want any further details or have any additional questions?

Here are 5 related questions to expand on this topic:

1. What is the difference between a relation and a function?
2. Can a function have multiple inputs mapping to the same output?
3. What is the vertical line test, and how is it related to determining functions graphically?
4. What happens if a domain value maps to multiple range values?
5. Can a range value have multiple domain values in a function?

Tip: When checking whether a relation is a function, always focus on the domain. Each domain element should map to only one range element.