Chapter 3 · Functions
Read f(input) as an output, substitute the complete input, and identify domain, rule and range.
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| Domain | x | Input | Formula calculation | Function output |
|---|---|---|---|---|
| — | — | — | — | — |
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Chapter 3 · Functions
Class ______ Name __________________ Date __________
In the projection model f(x) = x² + 1, switch between the finite domain {−2, −1, 0, 1, 2} and the real interval [−2, 2]. Predict whether 0.5 and 3 are allowed in each version.
In the finite-domain projection model f(x) = x² + 1, record the outputs at −2, −1, 0, 1 and 2, then write the attained range.
Test 0.5 and 3 in both model domains. Record the formula calculation separately from whether that input is allowed.
For the interval-domain model, record the minimum and maximum outputs and explain why the range is [1, 5], rather than just {1, 2, 5}.
Compare p(x) = x + 1 on {0, 1} and q(x) = x + 1 on [0, 1]. Give one input that distinguishes them, and write both ranges.
Let h(x) = 3x + 2 on {−1, 0, 2}. Find h(−1) and the range. Is h the same as k(x) = 3x + 2 on the reals?
Chapter 3 · Functions · Teacher reference
3.1.1(第2課時) · 02_讀懂函數符號與三要素.docx
5 min predict equal functions; 12 min compare mappings; 8 min organise notation; 12 min substitution practice; 3 min exit check.
In the projection model f(x) = x² + 1, switch between the finite domain {−2, −1, 0, 1, 2} and the real interval [−2, 2]. Predict whether 0.5 and 3 are allowed in each version.
In the projection model x² + 1, the finite domain gives range {1, 2, 5}; the real interval [−2, 2] gives [1, 5], since every output y in that interval is attained by x = √(y − 1). Input 0.5 is allowed only in the interval version; input 3 is allowed in neither, even though its formula calculation is 10. The paper examples use different rules: in f(a + 1) for f(x) = 2x − 3, a + 1 is the entire input. The paper’s two x² functions have the same range {0, 1} but different domains, so they are different functions. A function includes its domain and correspondence, which determine its range.
Compare p(x) = x + 1 on {0, 1} and q(x) = x + 1 on [0, 1]. Give one input that distinguishes them, and write both ranges.
Let h(x) = 3x + 2 on {−1, 0, 2}. Find h(−1) and the range. Is h the same as k(x) = 3x + 2 on the reals?
h(−1) = −1; the range is {−1, 2, 8}. They are different functions because their domains differ.
