Engaging students: Finding the domain and range of a function ... - Free Printable

It looks like you've uploaded a set of graphs labeled A through I (and possibly more, though only nine are visible), and the instruction says to use these graphs to answer questions 1–12, but the actual questions are missing.

Since the questions are not included in your message, I cannot solve them directly. However, I can help you analyze the graphs and explain what each one represents so that you can use this information to answer typical types of questions that might appear with such graphs.

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🔍 Overview of Each Graph

Let’s go through each graph and describe its key features:

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#### Graph A
- A piecewise linear function.
- Starts at (-4, 2), goes down to (-2, 1), up to (0, 3), down to (2, 1), then up to (4, 2).
- All points are closed circles, meaning all endpoints are included.
- Domain: [-4, 4]
- Range: [1, 3]
- It is not a function? Wait — let's check: for every x, only one y → yes, it is a function.
- Function? ✔ Yes
- Continuous? ✔ Yes (no breaks)

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#### Graph B
- A line segment from (-5, -2) to (3, 1), but open circle at (3,1).
- So it includes all points from x = -5 to x < 3.
- Domain: [-5, 3)
- Range: [-2, 1)
- Function? ✔ Yes (passes vertical line test)
- Continuous? ✔ Yes

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#### Graph C
- Vertical line segment from (-2, -3) to (-2, 3), open circles at both ends.
- So it's defined for x = -2, but not including y = -3 or y = 3.
- Domain: {-2}
- Range: (-3, 3)
- Function? ✘ No! Because for x = -2, there are multiple y-values → fails vertical line test.
- Continuous? ✔ Yes within its domain

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#### Graph D
- A circle centered at origin, radius ≈ 3 units.
- Equation: $x^2 + y^2 = 9$
- Domain: [-3, 3]
- Range: [-3, 3]
- Function? ✘ No! Fails vertical line test (e.g., x=0 gives two y-values)
- Continuous? ✔ Yes

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#### Graph E
- Two curves forming a "U" shape opening to the right.
- Looks like a parabola sideways, e.g., $x = y^2$
- Starts at (0, -2), goes to (4, -1), then to (4, 1), then to (0, 2)?
Wait — actually, it's symmetric about x-axis.
- Open circles at (0, -2) and (0, 2)? No — seems like closed at bottom and top?
Actually, endpoints: leftmost point is (0,0)? Wait, no:
- From (0, -2) to (4, 0) on upper branch? Actually, it appears:
- Bottom curve: from (0, -2) to (4, 0) to (4, 1)? No.
Let’s re-analyze:
- It's a sideways parabola: for each x ≥ 0, two y-values (except at vertex).
- The graph shows two branches: one going up from (0,0) to (4,2), and one going down from (0,0) to (4,-2).
But wait — it starts at (0,0) with an open circle?
No — actually, it has a closed dot at (0,0), and arrows at (4,2) and (4,-2) indicating continuation? But they're drawn as lines ending at (4,2) and (4,-2).

Wait — the graph shows:
- A V-shape opening to the right, starting at (0,0), going to (4,2) and (4,-2).
- But it's two lines: one from (0,0) to (4,2), and another from (0,0) to (4,-2).
- So it's like $x = |y|$ or something similar?

Actually, this is not a function because for x > 0, there are two y-values.

So:
- Domain: [0, 4]
- Range: [-2, 2]
- Function? ✘ No (fails vertical line test)
- Continuous? ✔ Yes

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#### Graph F
- Horizontal ray starting at (-2, 1) with an open circle, going to the right forever.
- So: $x > -2$, $y = 1$
- Domain: (-2, ∞)
- Range: {1}
- Function? ✔ Yes (only one output per input)
- Continuous? ✔ Yes

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#### Graph G
- Horizontal line segment from (-3, -1) to (2, -1), but open circle at (-3, -1), closed at (2, -1).
- So: $x \in (-3, 2]$, $y = -1$
- Domain: (-3, 2]
- Range: {-1}
- Function? ✔ Yes
- Continuous? ✔ Yes

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#### Graph H
- Line segment from (-3, 3) to (1, -1), closed circles at both ends.
- Domain: [-3, 1]
- Range: [-1, 3]
- Function? ✔ Yes
- Continuous? ✔ Yes

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#### Graph I
- V-shaped graph, like $y = |x|$, but with open circles at (-2,2) and (2,2), and closed circle at (0,0).
- So: $y = |x|$ for $x \in [-2, 2]$, but endpoints excluded.
- Domain: (-2, 2)
- Range: [0, 2)
- Function? ✔ Yes
- Continuous? ✔ Yes (within domain)

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🧠 Common Types of Questions That Might Appear

Since you said “answer questions 1–12”, here are likely question types and how to answer them using these graphs:

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🔹 Question Type 1: Which graph represents a function?

→ Use the vertical line test.
- Functions: A, B, F, G, H, I
- Not functions: C, D, E

> ✔ Answer: A, B, F, G, H, I

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🔹 Question Type 2: Which graph is continuous?

→ Check if there are jumps, holes, or breaks.
- All graphs except possibly those with open circles may be continuous if the open circles don’t break continuity.
- But open circles indicate holes, so:
- C: Has holes at top and bottom → discontinuous at endpoints
- B: Open at end → discontinuous at x=3
- F: Open at x=-2 → discontinuous at x=-2
- G: Open at x=-3 → discontinuous at x=-3
- I: Open at x=±2 → discontinuous at endpoints
- But continuity is defined on the domain. If the domain excludes those points, it can still be continuous on its domain.

So:
- Continuous on domain? All except maybe C and E?
- C: Vertical line, open ends → continuous on (-3,3) but not at endpoints → continuous on domain.
- E: Continuous on [0,4], even though it's not a function.
- D: Circle → continuous everywhere on domain.
- So most are continuous on their domains.

But if asking continuous over entire real line, then only ones without gaps are considered.

> Likely expected answers: A, D, H are continuous everywhere; others have restricted domains or holes.

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🔹 Question Type 3: What is the domain/range of graph X?

For example:
- Graph A: Domain: [-4,4], Range: [1,3]
- Graph B: Domain: [-5,3), Range: [-2,1)
- Graph C: Domain: {-2}, Range: (-3,3)
- Graph D: Domain: [-3,3], Range: [-3,3]
- Graph E: Domain: [0,4], Range: [-2,2]
- Graph F: Domain: (-2,∞), Range: {1}
- Graph G: Domain: (-3,2], Range: {-1}
- Graph H: Domain: [-3,1], Range: [-1,3]
- Graph I: Domain: (-2,2), Range: [0,2)

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🔹 Question Type 4: Which graph is NOT a function?

→ C, D, E

> ✔ Answer: C, D, E

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🔹 Question Type 5: Which graph has a domain of (-2, 2)?

→ Graph I

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🔹 Question Type 6: Which graph is a constant function?

→ F and G are constant functions (horizontal lines)
- F: y = 1, G: y = -1

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🔹 Question Type 7: Which graph passes through the origin?

→ A (yes, (0,3)), B (no), C (no), D (yes), E (yes, (0,0)), F (no), G (no), H (no), I (yes)
- So: A, D, E, I

But check:
- A: (0,3) → yes
- D: (0,3) → yes
- E: (0,0) → yes
- I: (0,0) → yes

✔ A, D, E, I

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🔹 Question Type 8: Which graph has a range of {1}?

→ F

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🔹 Question Type 9: Which graph is symmetric about the y-axis?

→ D (circle), I (V-shape), possibly A?

A: Not symmetric about y-axis (left side ≠ right side)
I: Yes — symmetric about y-axis
D: Yes — circle is symmetric
E: Symmetric about x-axis? Yes, but not y-axis
H: Not symmetric
G: Not symmetric
F: Not symmetric

✔ D, I

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🔹 Question Type 10: Which graph has a hole?

→ B (open at end), C (open at ends), F (open at start), G (open at start), I (open at ends)

So: B, C, F, G, I

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🔹 Question Type 11: Which graph is bounded?

→ Bounded means limited in size.
- A: bounded
- B: bounded
- C: bounded
- D: bounded (circle)
- E: bounded
- F: unbounded (goes to infinity)
- G: bounded
- H: bounded
- I: bounded

Only F is unbounded → so others are bounded.

✔ All except F

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🔹 Question Type 12: Which graph is increasing on its entire domain?

- A: increases then decreases → no
- B: increasing → yes
- C: vertical → not increasing
- D: not monotonic
- E: two branches → not increasing
- F: constant → not strictly increasing
- G: constant → no
- H: decreasing → no
- I: decreases then increases → no

Only B is increasing on its domain.

✔ B

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✔ Summary Table

| Graph | Function? | Continuous? | Domain | Range | Notes |
|-------|-----------|-------------|--------|-------|-------|
| A | ✔ | ✔ | [-4,4] | [1,3] | Piecewise |
| B | ✔ | ✔ | [-5,3) | [-2,1) | Increasing |
| C | ✘ | ✔ | {-2} | (-3,3) | Vertical line |
| D | ✘ | ✔ | [-3,3] | [-3,3] | Circle |
| E | ✘ | ✔ | [0,4] | [-2,2] | Sideways U |
| F | ✔ | ✔ | (-2,∞) | {1} | Ray |
| G | ✔ | ✔ | (-3,2] | {-1} | Horizontal |
| H | ✔ | ✔ | [-3,1] | [-1,3] | Line |
| I | ✔ | ✔ | (-2,2) | [0,2) | V-shape |

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📌 Final Note:

Please provide the actual questions 1–12, and I’ll give you exact answers based on the graphs.

But with this analysis, you can now answer most standard questions involving:
- Functions vs. relations
- Domain and range
- Continuity
- Symmetry
- Increasing/decreasing
- Boundedness
- etc.

Let me know what the specific questions are, and I’ll solve them step by step!