g10-05-11 · Core topics
Five powers, different forbidden inputs
Which of x, x², x³, √x and 1/x are defined at −1 and at 0?
1 MIN · PREDICT ON PAPER
On paper, mark allowed/rejected before matching each curve to its formula.
Discuss or write first. No typed answer is required to continue.
Probe a = -1; b = 2. 0/5 cards placed. The window shows only part of each graph.
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
Real √x requires x≥0; 1/x requires x≠0. The integer positive powers x, x² and x³ accept every real input.
Conditions: These are the five specified real-valued rules, not a claim that all real powers have the same domain.
Mathematical justification: Apply each operation’s real-domain condition. Polynomial integer powers require no divisions or roots; √x and 1/x do.
1 MIN · EXIT CHECK
List the five outputs at x=0, using “undefined” where needed.
Check after answering
0, 0, 0, 0, undefined, in the given order.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Use the zero case to distinguish “output zero” from “undefined”.
Watch for: Every power function is defined on all real numbers.
Challenge: Explain why x² and x³ agree at 0 and 1 but disagree at −1.
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.
Complete backup and restore