g10-05-13 · Core topics
Undo the input rule in the right order
If f(2x+1)=6x+5, how do you obtain f(u) rather than f((u−1)/2)?
1 MIN · PREDICT ON PAPER
For u=3, place “subtract 1” and “halve” in the order that recovers x.
Discuss or write first. No typed answer is required to continue.
u = 3. Your cards propose how to obtain x from the whole input 2x + 1.
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
Set u=2x+1, so x=(u−1)/2. Substitution yields f(u)=6(u−1)/2+5=3u+2. Undoing the input is not inverting f.
Conditions: The affine input rule is invertible over real numbers. In the discount context, eligibility is checked using the original price.
Mathematical justification: Substitute x=(u−1)/2 into the stated identity; then 2x+1=u exactly. For prices, 0.9p−20 differs from 0.9(p−20) by −2.
1 MIN · EXIT CHECK
From f(2x+1)=6x+5, find f(−3).
Check after answering
x=−2, hence f(−3)=−12+5=−7.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: After each rearrangement, run the recovered x forward through 2x+1 and check whether it returns u.
Watch for: Halve then subtract 1 is the inverse of doubling then adding 1.
Challenge: Compare “10% off then subtract 20” with the reverse order at an eligible original price of 100.
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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