Math Problem Statement

Determine which of these relations are functions: Relation 1 (mapping diagram), Relation 2 (table), Relation 3 (graph), Relation 4 (ordered pairs). Select either Function or Not a function next to each relation.

Solution

To determine whether the given relations are functions, we use the definition of a function:

A relation is a function if each input (x-value) maps to exactly one output (y-value).

Let’s analyze each relation one by one:

Relation 1: Diagram with ovals

  • From the diagram, the inputs xx are {1,2,3,6}\{1, 2, 3, 6\}.
  • Each input maps to exactly one output in the second oval:
    • 1→y11 \to y_1
    • 2→y32 \to y_3
    • 3→y53 \to y_5
    • 6→y76 \to y_7.
  • No input is repeated or mapped to multiple outputs.

✅ Relation 1 is a function.

Relation 2: Table

The table is:

x & : y \\ -2 & : 5 \\ 1 & : 5 \\ 2 & : 1 \\ 2 & : 5 \end{aligned}$$ - Look at the $$x$$-values: $$-2, 1, 2, 2$$. - Notice that $$x = 2$$ maps to **two different outputs** $$y = 1$$ and $$y = 5$$. ❌ **Relation 2 is NOT a function.** --- ### **Relation 3: Graph** - In the graph, we use the **vertical line test**: - If any vertical line intersects the graph at more than one point, it is **not a function**. - From the graph, a vertical line intersects the curve at multiple points (indicating multiple $$y$$-values for a single $$x$$-value). ❌ **Relation 3 is NOT a function.** --- ### **Relation 4: Set of ordered pairs** The pairs are: $$\{(-2, 1), (-2, 0), (0, 2), (1, 1), (2, -1)\}$$ - Check the $$x$$-values: $$-2, -2, 0, 1, 2$$. - Notice that $$x = -2$$ maps to **two different $$y$$-values**: $$1$$ and $$0$$. ❌ **Relation 4 is NOT a function.** --- ### Final Answers: 1. **Relation 1**: Function ✅ 2. **Relation 2**: Not a function ❌ 3. **Relation 3**: Not a function ❌ 4. **Relation 4**: Not a function ❌ Let me know if you need further clarification! --- ### Related Questions: 1. What is the **vertical line test** and how does it help identify functions? 2. Can a function have repeated $$y$$-values but unique $$x$$-values? Explain. 3. How can you represent relations using mapping diagrams or graphs? 4. What is the difference between a **relation** and a **function**? 5. How do ordered pairs help determine if a relation is a function? **Tip**: Remember, in a function, each input $$x$$ can only have one output $$y$$.

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Math Problem Analysis

Mathematical Concepts

Functions
Relations
Mapping Diagrams
Ordered Pairs
Graphs
Vertical Line Test

Formulas

-

Theorems

Vertical Line Test

Suitable Grade Level

Grades 8-10

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