g10-05-02 · Core topics
Same formula: the same function?
Does x = 0.5 distinguish a finite domain from the interval [−2,2]?
1 MIN · PREDICT ON PAPER
Predict whether 0.5 is allowed before dropping its chip onto each domain.
Discuss or write first. No typed answer is required to continue.
Same fixed rule x² + 1. First range: {1, 2, 5}. Second range: [1, 5].
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
Formula arithmetic and defined function output are different questions. For x²+1, 0.5 gives arithmetic 1.25, but this is a function value only when 0.5 belongs to the domain.
Conditions: Compare both the rule and the complete input domain. The range contains attained outputs only.
Mathematical justification: Exhibit one input contained in exactly one domain to refute equality. For a finite domain, evaluate every input to obtain its range.
1 MIN · EXIT CHECK
Find the range of x²+1 on {−2,−1,0,1,2}.
Check after answering
{1,2,5}; repeated outputs are listed once.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Require two sentences: “the arithmetic is …” and “the input is allowed/rejected because …”.
Watch for: The same expression defines the same function on any domain.
Challenge: For the two finite x² domains, explain how equal ranges {0,1} can coexist with different functions.
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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