Mrs. Newells Math: Relations and Functions Card Sort - Free Printable

Let's go through each of the four problems on the worksheet and determine whether each relation is a function.

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🔹 Definition of a Function:

A relation is a function if each input (x-value) corresponds to exactly one output (y-value).
In other words:
- No x-value can be paired with more than one y-value.
- Each x-value must have only one y-value.

We'll use this rule to analyze each problem.

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✔ Problem 1: Table

| x | -1 | 0 | 3 | 4 | -1 |
|---|----|---|---|---|----|
| y | -2 | 0 | -1 | -1 | 4 |

Step-by-step analysis:
- Look at the x-values: -1, 0, 3, 4, -1
- Notice that x = -1 appears twice, once with y = -2 and again with y = 4.
- So, one input (-1) has two different outputs (-2 and 4).

➡️ This violates the definition of a function.

Answer: ✘ Not a function

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✔ Problem 2: Graph

This is a V-shaped graph centered at the origin, opening upward — it looks like an absolute value graph: $ y = |x| $

Use the Vertical Line Test:
- Draw vertical lines across the graph.
- If any vertical line intersects the graph in more than one point, it's not a function.

But here:
- Every vertical line will intersect the graph in only one point.
- Even at x = 0, it touches just once.

✔ The graph passes the vertical line test.

Answer: ✔ Yes, it is a function

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✔ Problem 3: Mapping Diagram

We have:
- Left circle (inputs): -2, -1, -2
- Right circle (outputs): 9, 1, 2
- Arrows:
- -2 → 9
- -1 → 1
- -2 → 2

Wait: -2 appears twice as an input:
- Once pointing to 9
- Once pointing to 2

So, the input -2 maps to two different outputs: 9 and 2.

➡️ One input has two outputs → violates function rule

Answer: ✘ Not a function

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✔ Problem 4: Set of Ordered Pairs

Given:
$$
\{(-2, 1), (-3, 1), (-4, 1), (-5, 1)\}
$$

Check each x-value:
- -2 → 1
- -3 → 1
- -4 → 1
- -5 → 1

Each x-value is unique and maps to only one y-value (all map to 1, but that’s fine).

Even though all y-values are the same, that doesn't matter. What matters is that no x-value repeats.

✔ All inputs are distinct → this is a function

Answer: ✔ Yes, it is a function

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✔ Final Answers:

1. ✘ Not a function
2. ✔ Yes, it is a function
3. ✘ Not a function
4. ✔ Yes, it is a function

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📝 Summary:

| Problem | Is it a function? |
|--------|-------------------|
| 1 | No |
| 2 | Yes |
| 3 | No |
| 4 | Yes |

Let me know if you'd like explanations in simpler terms or visual help!