To determine if each relation is a function, let's analyze them individually. Remember, a function from a set $A$ (domain) to a set $B$ (range) is a relation in which each element of $A$ is associated with exactly one element of $B$.

Relation 1

In Relation 1, each element in the domain (door, paper, sun) has only one output in the range. There is no element in the domain that points to more than one element in the range.

• Conclusion: Relation 1 is a function.

Relation 2

In Relation 2, the element "5" in the domain is mapped to multiple values in the range (2, 5, -4, 3, and -6). Since a single input (5) has multiple outputs, it does not meet the criteria of a function.

• Conclusion: Relation 2 is not a function.

Relation 3

In Relation 3, we have the pairs $(d, w), (c, c), (w, c), (f, c)$. Each element in the domain (d, c, w, f) is mapped to only one output in the range. Even though multiple domain values map to "c" in the range, this does not violate the function rule.

• Conclusion: Relation 3 is a function.

Relation 4

In Relation 4, we have the pairs $(0, a), (-6, a), (4, a), (-6, j)$. Here, the element "-6" in the domain is associated with two different outputs, "a" and "j." This violates the definition of a function, where each input should map to exactly one output.

• Conclusion: Relation 4 is not a function.

Summary of Answers

• Relation 1: Function
• Relation 2: Not a function
• Relation 3: Function
• Relation 4: Not a function

Would you like further details on any specific part of this solution, or do you have any questions?

Here are five questions that might further deepen your understanding:

1. What are some examples of functions and non-functions in real-life situations?
2. How can you verify if a relation represented as a graph is a function?
3. Why does having multiple outputs for a single input violate the definition of a function?
4. Can a function have the same output for different inputs?
5. How would you represent a function or non-function using mapping diagrams?

Tip: Remember, a function can have the same range value for multiple domain values, but each domain value must map to only one range value.