Let's analyze each of the given relations to determine if they are functions:

Relation 1:

• Domain: pen, rock, lake

• Range: 3, 2, -2, 6, 8

• Each element in the domain maps to exactly one value in the range.

  • "pen" maps to both 3 and 2.
  • "rock" maps to both -2 and 6.

• This violates the definition of a function, where each element in the domain must map to only one value in the range.

  Answer: Relation 1 is not a function.

Relation 2:

• Domain: -2

• Range: 6, 2, -2

• The element -2 in the domain is mapped to three different values: 6, 2, and -2.

• Since a function requires each input (domain element) to have exactly one output (range value), this relation is not a function.

  Answer: Relation 2 is not a function.

Relation 3:

• Given as: $\{(c, c), (d, c), (e, d), (n, c)\}$

• Each element in the domain ($c, d, e, n$) is mapped to a unique element in the range. No domain element is mapped to more than one range element.

  Answer: Relation 3 is a function.

Relation 4:

• Given as: $\{(-8, s), (9, s), (5, v), (5, s)\}$

• The element 5 in the domain is mapped to two different values: $v$ and $s$.

• Since this violates the rule that a domain element must map to exactly one range value, this relation is not a function.

  Answer: Relation 4 is not a function.

Summary:

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

Let me know if you'd like further clarification or a detailed explanation!

Here are five related questions to deepen your understanding:

1. What is the definition of a function in mathematics?
2. Can a function have two domain elements mapping to the same range value?
3. How can a relation fail to be a function if it is given graphically?
4. What is the difference between a surjective and injective function?
5. How can you identify a function from a set of ordered pairs?

Tip: A function is valid if every element in the domain has exactly one corresponding element in the range, but the reverse is not necessary—multiple domain elements can map to the same range element!