05 Functions and their properties · g10-05-01 · 5–10 min
Repair one mapping arrow
Can two inputs share an output? Which single input breaks the function rule?
Predict: Before changing the diagram, mark each input with 0, 1 or 2 distinct outputs.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
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| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: All three inputs map to 8. Function or not, and why?
05 Functions and their properties · g10-05-02 · 5–10 min
Same formula: the same function?
Does x = 0.5 distinguish a finite domain from the interval [−2,2]?
Predict: Predict whether 0.5 is allowed before dropping its chip onto each domain.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: Find the range of x²+1 on {−2,−1,0,1,2}.
05 Functions and their properties · g10-05-03 · 5–10 min
Which endpoint belongs?
Why is 3 allowed in √(x−3), while 2 is forbidden in 1/(x−2)?
Predict: Draw the boundary point as open or closed before selecting each formula.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
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| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: Write the domain of 1/√(x−3).
05 Functions and their properties · g10-05-04 · 5–10 min
Keep every restriction after simplification
Does cancelling x−1 restore the missing input x=1?
Predict: Predict whether the original and simplified formulas both have value 2 at x=1.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: Is (1,2) on the original graph of (x²−1)/(x−1)?
05 Functions and their properties · g10-05-05 · 5–10 min
Should the points be joined?
The same cost formula y=2x+4 describes time or item count. Which graph permits x=2.5?
Predict: Draw five table points. Predict whether connecting them preserves the intended inputs.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
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| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: For 0–4 whole tickets costing 2x+4, is (2.5,9) on the graph?
05 Functions and their properties · g10-05-06 · 5–10 min
A missing boundary or two outputs?
If both pieces include the boundary, is the relation always invalid?
Predict: At t=1, count distinct outputs for the open, single-closed and double-closed versions.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: Both formulas give 6 at t=1 and both include it. How many function values occur?
05 Functions and their properties · g10-05-07 · 5–10 min
Swap two function machines
Does doubling then adding 3 give the same result as adding 3 then doubling?
Predict: For input 2, predict both intermediate values and final outputs before sending the token.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: At x=0, which is defined: √(x−1) or √x−1?
05 Functions and their properties · g10-05-08 · 5–10 min
Can the largest value be reached?
Does y=x on (0,2) have a maximum just because the graph approaches 2?
Predict: Write a candidate maximum; predict what happens when the endpoint is closed.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
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| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: State maximum and minimum of y=x on (0,2].
05 Functions and their properties · g10-05-09 · 5–10 min
Decreasing on each piece or the whole domain?
Is 1/x decreasing on all nonzero real numbers?
Predict: Predict the order of f(−1) and f(1), then compare it with the order of the inputs.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: Give one pair refuting whole-domain decrease of 1/x.
05 Functions and their properties · g10-05-10 · 5–10 min
Symmetry needs the matching input
Is x² still even when its domain is restricted to [0,3]?
Predict: For x=2, first ask whether −2 is allowed; only then compare f(−x) and f(x).
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: Can a function be both odd and even on a symmetric nonempty domain?
05 Functions and their properties · g10-05-11 · 5–10 min
Five powers, different forbidden inputs
Which of x, x², x³, √x and 1/x are defined at −1 and at 0?
Predict: On paper, mark allowed/rejected before matching each curve to its formula.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
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| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: List the five outputs at x=0, using “undefined” where needed.
05 Functions and their properties · g10-05-12 · 5–10 min
When does the larger exponent give less?
For positive x, does x³ always exceed x²?
Predict: Predict the ordering for x=0.5, 1 and 2 before moving the probe.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: Compare 0.8² and 0.8³ without decimal multiplication.
05 Functions and their properties · g10-05-13 · 5–10 min
Undo the input rule in the right order
If f(2x+1)=6x+5, how do you obtain f(u) rather than f((u−1)/2)?
Predict: For u=3, place “subtract 1” and “halve” in the order that recovers x.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 3 | |
Revise: My claim needs the condition… because…
Exit: From f(2x+1)=6x+5, find f(−3).
05 Functions and their properties · g10-05-14 · 5–10 min
Cheapest at this quantity or most within budget?
Can the best plan change when the decision question changes?
Predict: At n=10, 30 and 50, predict the cheaper plan; then use budget 160 to predict the greatest affordable count.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
| Case 1 | |
| Case 2 | |
| Case 4 | |
Revise: My claim needs the condition… because…
Exit: With budget 160, why is P limited to 46 rather than 47?
05 Functions and their properties · g10-05-15 · 5–10 min
Squaring can create a false root
Why does squaring √(x+2)=x produce two candidates but only one solution?
Predict: Predict whether x=−1 and x=2 satisfy the original equation before checking either graph.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Test input x = 2 | |
Case 2
Test input x = -1 | |
Case 3
Test input x = -3 | |
Revise: My claim needs the condition… because…
Exit: Solve √(x+2)=x and identify the extraneous candidate.
