g10-05-10 · Core topics
Symmetry needs the matching input
Is x² still even when its domain is restricted to [0,3]?
1 MIN · PREDICT ON PAPER
For x=2, first ask whether −2 is allowed; only then compare f(−x) and f(x).
Discuss or write first. No typed answer is required to continue.
Domain ℝ. f(2) = 4; actual f(-2) = 4.
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
Odd/even definitions require a domain symmetric about zero. x² on [0,3] fails that requirement. On a symmetric domain, x² is even, x³ odd, and the zero function both.
Conditions: Check domain symmetry before the identities f(−x)=f(x) or f(−x)=−f(x).
Mathematical justification: On a symmetric domain, (−x)²=x², (−x)³=−x³, and 0=−0. The restricted interval lacks −2 while containing 2.
1 MIN · EXIT CHECK
Can a function be both odd and even on a symmetric nonempty domain?
Check after answering
Yes: f(x)=0. If both identities hold, f(x)=−f(x), so every value is zero.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Ask for the domain test and identity test as two separate decisions.
Watch for: The formula alone decides parity; odd and even are mutually exclusive.
Challenge: Compare the mirrored point with the actual value for x+1. Identify a numerical failure of evenness.
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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