g10-05-03 · Core topics
Which endpoint belongs?
Why is 3 allowed in √(x−3), while 2 is forbidden in 1/(x−2)?
1 MIN · PREDICT ON PAPER
Draw the boundary point as open or closed before selecting each formula.
Discuss or write first. No typed answer is required to continue.
Painted set: [0, 4). Infinity is a direction and always uses an open bracket.
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
A real square root permits zero radicand, giving x ≥ 3. A denominator cannot be zero, giving x ≠ 2. In [0,4), 0 is included and 4 excluded.
Conditions: Work over real inputs. Infinity is a direction, never an included endpoint; integers and real intervals are different domains.
Mathematical justification: Translate the defining restrictions: x−3 ≥ 0 and x−2 ≠ 0. These algebraic conditions justify all inputs; plotted examples only illustrate them.
1 MIN · EXIT CHECK
Write the domain of 1/√(x−3).
Check after answering
(3,∞): the radicand must be positive because the root is also a denominator.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Ask “what operation fails here?” before naming an interval bracket.
Watch for: Every domain boundary must be excluded.
Challenge: Toggle integer inputs: identify a noninteger that leaves the domain even though it lies between the endpoints.
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
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