g10-05-06 · Core topics
A missing boundary or two outputs?
If both pieces include the boundary, is the relation always invalid?
1 MIN · PREDICT ON PAPER
At t=1, count distinct outputs for the open, single-closed and double-closed versions.
Discuss or write first. No typed answer is required to continue.
Fixed total domain: 0 < t ≤ 4. First branch 0 < t ≤ 1; second branch t > 1, t ≤ 4. At 1: 1 applicable branch(es), 1 distinct output(s).
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Both open endpoints omit the boundary. Both closed endpoints are harmless if both formulas give the same value there. An overlap giving two distinct values violates uniqueness.
Conditions: For the rental model, the values 6 and 6+4(t−1) agree at t=1. Equality at one point does not justify an interval of overlap.
Mathematical justification: At a shared input, form the set of distinct outputs. A function requires cardinality exactly one. The four constructions exhibit zero, one and two outputs.
1 MIN · EXIT CHECK
Both formulas give 6 at t=1 and both include it. How many function values occur?
Check after answering
One distinct value, 6.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Have students state both formulas at the boundary before judging the brackets.
Watch for: A function must never have overlapping piece conditions, even with identical outputs.
Challenge: Use the paper model x+2 and x² at x=0 to explain why closing both ends now fails.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
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