Handwritten solutions to algebra problems involving function domains, ranges, and continuity, displayed on a classroom worksheet.

You've uploaded a worksheet titled “Functions and Continuity” from *Reveal Algebra 2*, and you’re asking to solve the problem and explain the solution.

Since the worksheet contains multiple problems (Examples 4: #9–11; Mixed Exercises: #12–17), I’ll go through each graph one by one, correct any errors in the handwritten answers, and provide clear explanations for domain, range, and continuity using both set-builder notation and interval notation.

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## ✔ General Definitions:

- Domain: All possible input values (x-values) for which the function is defined.
- Range: All possible output values (y-values) the function can produce.
- Set-builder notation: e.g., `{x | x ∈ ℝ}` or `{y | y ≥ -8}`
- Interval notation: e.g., `(-∞, ∞)` or `[-8, ∞)`
- Discrete: Graph consists of separate points (not connected).
- Continuous: Graph is an unbroken curve/line — no gaps or jumps.
- Neither: Not discrete (has connected parts) but also not fully continuous (e.g., has holes or asymptotes).

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# 🔢 Examples 4

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📌 Problem 9:

Graph: A straight line going diagonally up-right (like y = x - 4). Extends infinitely in both directions.

✔ Correct Answer:
- Domain: All real numbers → `D = {x | x ∈ ℝ}` or `(-∞, ∞)`
- Range: All real numbers → `R = {y | y ∈ ℝ}` or `(-∞, ∞)`
- Continuity: Continuous (straight line with no breaks)

✘ Handwritten answer says range is `(-∞, 8)` — incorrect. The line goes infinitely high and low.

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📌 Problem 10:

Graph: Downward-opening parabola with vertex at (0,0). Extends infinitely downward.

✔ Correct Answer:
- Domain: All real numbers → `D = {x | x ∈ ℝ}` or `(-∞, ∞)`
- Range: y-values ≤ 0 → `R = {y | y ≤ 0}` or `(-∞, 0]`
- Continuity: Continuous (smooth curve)

✔ Handwritten answer is correct.

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📌 Problem 11:

Graph: S-shaped curve (like a cubic function or sideways sine wave) that passes through origin and extends infinitely in all directions.

✔ Correct Answer:
- Domain: All real numbers → `D = {x | x ∈ ℝ}` or `(-∞, ∞)`
- Range: All real numbers → `R = {y | y ∈ ℝ}` or `(-∞, ∞)`
- Continuity: Continuous (no breaks)

✘ Handwritten answer says domain is `[-2, 2]` — wrong. The graph clearly extends beyond x = ±2.

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# 🧩 Mixed Exercises (12–17)

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📌 Problem 12:

Graph: Upward-opening parabola with vertex at (0, -8). Extends infinitely upward.

✔ Correct Answer:
- Domain: All real numbers → `D = {x | x ∈ ℝ}` or `(-∞, ∞)`
- Range: y ≥ -8 → `R = {y | y ≥ -8}` or `[-8, ∞)`
- Continuity: Continuous

✔ Handwritten answer is mostly correct, though notation is messy. Range should be `[-8, ∞)`, not `(-8, ∞)`.

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📌 Problem 13:

Graph: W-shaped curve (like a quartic polynomial). Has local minima and maxima. Extends infinitely left/right and up/down.

✔ Correct Answer:
- Domain: All real numbers → `D = {x | x ∈ ℝ}` or `(-∞, ∞)`
- Range: y ≥ -4 (lowest point is at y = -4) → `R = {y | y ≥ -4}` or `[-4, ∞)`
- Continuity: Continuous

✔ Handwritten answer is correct.

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📌 Problem 14:

Graph: Cubic-like curve passing through origin, increasing steeply on both sides.

✔ Correct Answer:
- Domain: All real numbers → `D = {x | x ∈ ℝ}` or `(-∞, ∞)`
- Range: All real numbers → `R = {y | y ∈ ℝ}` or `(-∞, ∞)`
- Continuity: Continuous

✔ Handwritten answer says “Continuous” — correct. Domain/range not written, but implied.

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📌 Problem 15:

Graph: Straight line going down-right (like y = -x + b). Extends infinitely.

✔ Correct Answer:
- Domain: All real numbers → `D = {x | x ∈ ℝ}` or `(-∞, ∞)`
- Range: All real numbers → `R = {y | y ∈ ℝ}` or `(-∞, ∞)`
- Continuity: Continuous

✔ Handwritten answer is correct.

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📌 Problem 16:

Graph: V-shaped absolute value graph (vertex at origin). Opens upward.

✔ Correct Answer:
- Domain: All real numbers → `D = {x | x ∈ ℝ}` or `(-∞, ∞)`
- Range: y ≥ 0 → `R = {y | y ≥ 0}` or `[0, ∞)`
- Continuity: Continuous

✘ Handwritten answer says range is `(-∞, ∞)` — incorrect. The graph never goes below y=0.

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📌 Problem 17:

Graph: Wavy curve, starts low left, rises, dips slightly, then rises again. Extends infinitely left/right and up/down.

✔ Correct Answer:
- Domain: All real numbers → `D = {x | x ∈ ℝ}` or `(-∞, ∞)`
- Range: All real numbers → `R = {y | y ∈ ℝ}` or `(-∞, ∞)`
- Continuity: Continuous

✔ Handwritten answer is correct.

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# 📝 Summary Table:

| Problem | Domain (Set-Builder) | Domain (Interval) | Range (Set-Builder) | Range (Interval) | Continuous? |
|---------|------------------------------|-------------------|--------------------------|------------------|-------------|
| 9 | `{x | x ∈ ℝ}` | `(-∞, ∞)` | `{y | y ∈ ℝ}` | `(-∞, ∞)` | ✔ Yes |
| 10 | `{x | x ∈ ℝ}` | `(-∞, ∞)` | `{y | y ≤ 0}` | `(-∞, 0]` | ✔ Yes |
| 11 | `{x | x ∈ ℝ}` | `(-∞, ∞)` | `{y | y ∈ ℝ}` | `(-∞, ∞)` | ✔ Yes |
| 12 | `{x | x ∈ ℝ}` | `(-∞, ∞)` | `{y | y ≥ -8}` | `[-8, ∞)` | ✔ Yes |
| 13 | `{x | x ∈ ℝ}` | `(-∞, ∞)` | `{y | y ≥ -4}` | `[-4, ∞)` | ✔ Yes |
| 14 | `{x | x ∈ ℝ}` | `(-∞, ∞)` | `{y | y ∈ ℝ}` | `(-∞, ∞)` | ✔ Yes |
| 15 | `{x | x ∈ ℝ}` | `(-∞, ∞)` | `{y | y ∈ ℝ}` | `(-∞, ∞)` | ✔ Yes |
| 16 | `{x | x ∈ ℝ}` | `(-∞, ∞)` | `{y | y ≥ 0}` | `[0, ∞)` | ✔ Yes |
| 17 | `{x | x ∈ ℝ}` | `(-∞, ∞)` | `{y | y ∈ ℝ}` | `(-∞, ∞)` | ✔ Yes |

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# 🎯 Key Takeaways:

- If a graph extends infinitely left/right → domain is all real numbers.
- If it extends infinitely up/down → range is all real numbers.
- If it has a lowest/highest point → range is bounded below/above.
- All graphs here are continuous — none have holes, jumps, or asymptotes.
- Always check if your range includes the minimum/maximum value (use brackets `[ ]` if included).

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✔ Final Note: You did well overall! Just watch out for misreading graphs (like #9, #11, #16) and always double-check whether endpoints are included (`[ ]`) or excluded (`( )`).

Let me know if you want this as a printable answer key or need help with more problems!