https://file.aiquickdraw.com/mathbot/file/1727303216973t7cmbr0k.jpg

Let's analyze each relation to determine whether it's a function or not.

### Definition of a Function:
A relation is a **function** if every element in the domain is mapped to exactly one element in the range. In other words, no input from the domain has more than one output in the range.

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### **Relation 1:**
- **Domain:** \{t, h, e, d\}
- **Range:** \{1, -9, -7\}
- The arrows indicate that:
  - t → 1
  - h → -9
  - e → -9
  - d → -7

Since no domain element is mapped to more than one range element, **Relation 1 is a function**.

---

### **Relation 2:**
- **Domain:** \{y, d, k, h\}
- **Range:** \{d, k, h, u, y\}
- The arrows indicate that:
  - y → d
  - d → k
  - k → h
  - h → y
  - h → u (Note: h maps to both u and y)

Since the domain element "h" maps to two different range elements (u and y), **Relation 2 is not a function**.

---

### **Relation 3:**
- **Domain:** \{v, z, s, n, r\}
- **Range:** \{pen\}

All domain elements map to a single range element, "pen." Since each domain element maps to only one value in the range, **Relation 3 is a function**.

---

### **Relation 4:**
- **Domain:** \{-8, 9\}
- **Range:** \{3, -2, 9, 6, 5\}
- The table shows:
  - -8 → 3
  - 9 → -2
  - 9 → 9
  - 9 → 6
  - -8 → 5

Since both domain elements (-8 and 9) map to more than one range element (-8 maps to 3 and 5, and 9 maps to -2, 9, and 6), **Relation 4 is not a function**.

---

### Final Classification:
1. **Relation 1:** Function
2. **Relation 2:** Not a function
3. **Relation 3:** Function
4. **Relation 4:** Not a function

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Let me know if you'd like any further details or have questions! Here are 5 questions for further practice:

1. How would you modify Relation 4 to make it a function?
2. Can a relation with a constant range value ever not be a function?
3. What is the key difference between Relation 2 and Relation 3 in terms of their mappings?
4. How can you visually identify a function from a mapping diagram?
5. Why does Relation 2 fail to be a function, even though each domain element appears only once?

**Tip:** A function can have the same range value for different domain values, but each domain value must map to exactly one range value.

Based on the given relations, determine whether each relation represents a function or not.

Learn how to determine whether a relation is a function by analyzing mapping diagrams. This lesson covers key concepts about functions and relations and provides a step-by-step solution for identifying functions based on domain and range mappings.

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