To determine which of the given relations are functions, we need to check if each relation assigns exactly one output (i.e., $y$ value) for each input (i.e., $x$ value). A relation is not a function if any $x$ value corresponds to more than one $y$ value.

Let's analyze each relation:

1. Relation 1: $\{(41, -15), (18, 32), (90, 14), (41, 14)\}$

   • Here, the $x$-value 41 corresponds to both $-15$ and 14.
   • This is not a function.

2. Relation 2: $\{(-84, 78), (28, 4), (69, 54), (-84, -70)\}$

   • Here, the $x$-value $-84$ corresponds to both 78 and $-70$.
   • This is not a function.

3. Relation 3: $\{(-13, 14), (-66, -35), (-28, 14), (0, -28)\}$

   • All $x$-values are unique.
   • This is a function.

4. Relation 4: $\{(76, 92), (73, 76), (57, -8), (9, 20)\}$

   • All $x$-values are unique.
   • This is a function.

Summary:

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

Would you like more details or have any further questions?

Here are some related questions for further practice:

1. How do you define a function in terms of relations?
2. Can a function have the same $y$-value for different $x$-values?
3. What is the difference between a relation and a function?
4. How would you graphically determine if a relation is a function?
5. Can a function have more than one output for a single input?

Tip: Remember that a function must assign exactly one output for each input. If any input corresponds to more than one output, the relation is not a function.