Chapter 3 · Functions
Use arbitrary ordered inputs to prove monotonicity on a stated interval, and use one counterexample to refute an overbroad claim.
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| x₁ | x₂ | f(x₁) | f(x₂) | f(x₂) − f(x₁) |
|---|---|---|---|---|
| 1 | 2 | 1 | 0.5 | -0.5 |
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Chapter 3 · Functions
Class ______ Name __________________ Date __________
Predict whether 1/x is decreasing on its whole domain. Separate pairs on the same side of zero from a pair straddling zero.
Test 1/x at ordered pairs (−2, −1), (1, 2) and (−1, 1). Record whether the output order agrees with decreasing behaviour.
For f(x) = 2x² + 1, take arbitrary 0 < x₁ < x₂. Factor f(x₂) − f(x₁) and write the sign of every factor.
For −x² + 4x + 1, complete the square and record where the maximum is attained. Explain why a few sampled values cannot rule out an unseen larger value.
Prove that h(x) = 5 − 3x is strictly decreasing on the reals by comparing arbitrary x₁ < x₂. State the sign of the difference before giving the conclusion.
A student says “1/x decreases on its entire domain because each branch slopes down.” Give a counterexample and state the two correct decreasing intervals.
Chapter 3 · Functions · Teacher reference
3.2.1(第9課時) · 09_用定義說明單調性.docx
5 min predict the reciprocal claim; 12 min find a counterexample and factor differences; 8 min organise a definition proof; 12 min paper reasoning; 3 min exit check.
Predict whether 1/x is decreasing on its whole domain. Separate pairs on the same side of zero from a pair straddling zero.
Strict increase on an interval means that every pair x₁ < x₂ in that interval satisfies f(x₁) < f(x₂). Finite checks suggest a conjecture; they do not prove this universal statement. For 2x² + 1 and 0 < x₁ < x₂, the difference 2(x₂ − x₁)(x₂ + x₁) is positive, proving increase. For 1/x, the difference −(x₂ − x₁)/(x₁x₂) is negative when both inputs lie on the same side of zero. The pair −1 < 1 gives −1 < 1 in the outputs, refuting decrease across the entire punctured domain.
Prove that h(x) = 5 − 3x is strictly decreasing on the reals by comparing arbitrary x₁ < x₂. State the sign of the difference before giving the conclusion.
A student says “1/x decreases on its entire domain because each branch slopes down.” Give a counterexample and state the two correct decreasing intervals.
Take −1 < 1: f(−1) = −1 < f(1) = 1, contradicting decrease. The function is decreasing separately on (−∞, 0) and (0, ∞).
