g10-05-09 · Core topics
Decreasing on each piece or the whole domain?
Is 1/x decreasing on all nonzero real numbers?
1 MIN · PREDICT ON PAPER
Predict the order of f(−1) and f(1), then compare it with the order of the inputs.
Discuss or write first. No typed answer is required to continue.
x₁ = 1, x₂ = 2. f(x₂) − f(x₁) = -0.5.
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
1/x decreases separately on (−∞,0) and (0,∞). Across zero, −1<1 but f(−1)=−1<f(1)=1, which refutes decrease on the whole domain.
Conditions: For strict decrease, every allowed pair x₁<x₂ must satisfy f(x₁)>f(x₂). x=0 is not allowed.
Mathematical justification: For same-sign x₁<x₂, 1/x₁−1/x₂=(x₂−x₁)/(x₁x₂)>0. A crossing-zero pair has negative denominator and reverses this sign.
1 MIN · EXIT CHECK
Give one pair refuting whole-domain decrease of 1/x.
Check after answering
x₁=−1, x₂=1 gives −1<1 and f(x₁)<f(x₂).
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Keep x₁<x₂ visible and ask which word in “for every pair” the counterexample defeats.
Watch for: Two decreasing branches imply decrease on their union.
Challenge: For 2x²+1 on (0,∞), use a difference of squares to prove increase.
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.
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