Representing a Function Worksheet with solutions, displaying a mapping diagram, a circle, and a parabolic function graph.

It looks like you've uploaded a worksheet titled "Representing a Function Worksheet (with solutions)" that includes three different representations of relations or functions:

1. A mapping diagram (left side),
2. A graph of a circle (top right),
3. A graph of a cubic-like curve (bottom right).

Let’s analyze each one and determine whether it represents a function, and explain why.

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🔹 1. Mapping Diagram

Input values: {1, 2, 3, 4}
Output values: {8, 10, 12, 14, 16}

Arrows:
- 1 → 10
- 2 → 12
- 3 → 14
- 4 → 16

✔ Is this a function? YES

Explanation:
Each input (from the left set) maps to exactly one output. There are no inputs with multiple outputs, and no input is missing. This satisfies the definition of a function.

> Function Definition: A relation where each input has exactly one output.

Also, we can observe a pattern:
- 1 → 10 = 2×1 + 8?
Wait: Let's check:
- 1 → 10 → 10 = 2×5 → not obvious.
But look at the outputs: 10, 12, 14, 16 — increasing by 2.
Inputs: 1, 2, 3, 4 → so maybe:
- f(1) = 10
- f(2) = 12
- f(3) = 14
- f(4) = 16

So: f(x) = 2x + 8?

Check:
- 2(1)+8 = 10 ✔
- 2(2)+8 = 12 ✔
- 2(3)+8 = 14 ✔
- 2(4)+8 = 16 ✔

Yes! So the function is:
👉 f(x) = 2x + 8

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🔹 2. Graph of a Circle

This graph shows a circle centered at the origin with radius 3 (since it goes from -3 to 3 on both x and y axes).

Equation: $ x^2 + y^2 = 9 $

✘ Is this a function? NO

Explanation:
Use the Vertical Line Test:
- If any vertical line intersects the graph more than once, it's not a function.
- For example, draw a vertical line at x = 0: it hits the circle at y = 3 and y = -3 → two outputs for one input.

So, for a single x-value (like x=0), there are two y-values → violates the function rule.

➡️ Therefore, this is a relation, but not a function.

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🔹 3. Graph of a Cubic-Like Curve

This graph shows a smooth curve that:
- Crosses the x-axis at approximately x = -2, x = 0, and x = 2,
- Has a local maximum around (-1, 2),
- A local minimum around (1, -2),
- Continues upward as x increases.

✔ Is this a function? YES

Explanation:
Apply the Vertical Line Test:
- Any vertical line will intersect the graph only once.
- No matter where you draw a vertical line, it touches only one point.

Even though the graph curves up and down, it passes the test.

➡️ Therefore, this is a function.

It likely represents a cubic function, such as:
$$ f(x) = x^3 - x \quad \text{or similar} $$

But we don’t need the exact equation unless asked.

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✔ Final Summary

| Representation | Is it a Function? | Reason |
|----------------|-------------------|--------|
| Mapping Diagram | ✔ Yes | Each input maps to exactly one output |
| Circle Graph | ✘ No | Fails Vertical Line Test (e.g., x=0 gives two y-values) |
| Cubic Curve | ✔ Yes | Passes Vertical Line Test |

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📌 Key Concept Recap

A function must assign exactly one output to each input.

- Mapping diagrams: Check if any input points to more than one output.
- Graphs: Use the Vertical Line Test.

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Let me know if you'd like help writing equations for these or solving additional problems!