Chapter 3 · Functions
Substitute an entire expression into a rule, recover a rule by a reversible change of variable, and compare the order of two discounts.
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| Order | Intermediate amount | Final price |
|---|---|---|
| Student order | — | — |
| Reverse order | — | — |
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Chapter 3 · Functions
Class ______ Name __________________ Date __________
For an original price x ≥ 100, predict which discount order is cheaper. The threshold applies to the original price only; both operations may then use the intermediate amount.
Let D(u) = 0.9u and C(u) = u − 20. At original prices 100 and 200, record the intermediate and final amounts for C(D(x)) and D(C(x)).
Derive both formulas and subtract them. Record whether the saving depends on the original price under this model.
Given f(2x + 1) = 6x + 5 for every real x, set t = 2x + 1. Record x in terms of t and rewrite the entire identity.
Recover h(x) from h(3x − 2) = 6x + 1 for every real x. Then compare the two discount orders with a subtraction of 30 instead of 20; justify the difference algebraically.
An original price is 100 and qualifies for both operations under the original-price-only rule. Compare 10% off then subtract 20 with subtract 20 then 10% off. Explain the general saving.
Chapter 3 · Functions · Teacher reference
3.4.1(第13課時) · 13_複合規則與簡單換元.docx
5 min predict discount order; 12 min track prices and derive the difference; 8 min connect to substitution; 12 min paper composition and change of variable; 3 min exit check.
For an original price x ≥ 100, predict which discount order is cheaper. The threshold applies to the original price only; both operations may then use the intermediate amount.
Composition replaces the complete input of the outer rule. Here C(D(x)) = 0.9x − 20 and D(C(x)) = 0.9x − 18, so discounting first is 2 cheaper for every original x ≥ 100. The eligibility condition is on the original price only; it is not reapplied to intermediate amounts in this activity. To recover f from f(2x + 1), use the reversible substitution t = 2x + 1, hence x = (t − 1)/2. Since t covers all reals, f(t) = 3t + 2 determines the full real rule. If an identity were given only on a restricted x-domain, the recovered t-domain would also need to be restricted.
Recover h(x) from h(3x − 2) = 6x + 1 for every real x. Then compare the two discount orders with a subtraction of 30 instead of 20; justify the difference algebraically.
An original price is 100 and qualifies for both operations under the original-price-only rule. Compare 10% off then subtract 20 with subtract 20 then 10% off. Explain the general saving.
Discount then subtract: 90 − 20 = 70. Subtract then discount: 80 × 0.9 = 72. In general the formulas are 0.9x − 20 and 0.9x − 18, so the first order saves 2.
