Which of the following relations are functions?

{
(
41
,
−
15
)
,
(
18
,
32
)
,
(
90
,
14
)
,
(
41
,
14
)
}

{
(
−
84
,
78
)
,
(
28
,
4
)
,
(
69
,
54
)
,
(
−
84
,
−
70
)
}

{
(
−
13
,
14
)
,
(
−
66
,
−
35
)
,
(
−
28
,
14
)
,
(
0
,
−
28
)
}

{
(
76
,
92
)
,
(
73
,
76
)
,
(
57
,
−
8
)
,
(
9
,
20
)
}

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.

Learn how to identify functions among given relations. Understand the criteria that define a function in mathematics through examples and detailed analysis. Suitable for students in Grades 9-12.

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