Intersect simultaneous input restrictions, preserve excluded inputs after cancellation, and find attained outputs on a specified domain.
Combine restrictions and find ranges
Paper starting model
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Overlaid masks: 0/2. Current intersection accepts x = 2.
Give your reason before checking.
Numeric readings
Values in this design
x
Current mask intersection
Original function output
2
accepted
undefined
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Chapter 3 · Functions
Combining domain restrictions and finding ranges
Class ______ Name __________________ Date __________
How many gates must an input pass, and can simplifying the formula reopen a closed gate?
My prediction and reason
For √(x + 1)/(x − 2), predict whether −1 and 2 are allowed. For (x² − 1)/(x − 1), predict whether cancellation makes x = 1 valid.
Evidence and observations
On two number lines record x ≥ −1 and x ≠ 2, then shade their common part. Check −1, 0 and 2 explicitly.
Compare the original fraction (x² − 1)/(x − 1) with x + 1 at x = 0, 1, 2. Record where equality can be stated.
For x² on [−2, 1], record the endpoint values and the value at 0. Explain how every output from 0 to 4 is attained.
Explain or give a counterexample
Transfer
Find the domain of √(x + 2)/(x − 1). Then find the range of x² when the domain is {−2, 0, 1}; explain why it is not an interval.
Exit task
Find the domain of √(x + 1)/(x − 2). Explain whether (x² − 1)/(x − 1) and x + 1 are the same function on their natural real domains.
Chapter 3 · Functions · Teacher reference
Combining domain restrictions and finding ranges
3.1.1(第4課時) · 04_定義域的多重限制與值域.docx
5 min predict gate failures; 12 min overlay restrictions and compare the hole; 8 min organise the range argument; 12 min paper examples; 3 min exit check.
How many gates must an input pass, and can simplifying the formula reopen a closed gate?
For √(x + 1)/(x − 2), predict whether −1 and 2 are allowed. For (x² − 1)/(x − 1), predict whether cancellation makes x = 1 valid.
Evidence tasks
On two number lines record x ≥ −1 and x ≠ 2, then shade their common part. Check −1, 0 and 2 explicitly.
Compare the original fraction (x² − 1)/(x − 1) with x + 1 at x = 0, 1, 2. Record where equality can be stated.
For x² on [−2, 1], record the endpoint values and the value at 0. Explain how every output from 0 to 4 is attained.
Reference reasoning
An input must satisfy every restriction, so use intersection: √(x + 1)/(x − 2) has domain [−1, 2) ∪ (2, ∞). A square root in a denominator must be strictly positive. Cancellation preserves values only where the original expression is defined; the hole at x = 1 remains. For x² on [−2, 1], outputs are bounded by 0 and 4, and every y in [0, 4] is attained by x = −√y in [−2, 0]. Thus the range is exactly [0, 4].
Find the domain when a square root is in the denominator.
f(x)=x−11
The radicand requires x − 1 ≥ 0; the denominator must also be nonzero.
Together these give x − 1 > 0. The domain is (1, ∞).
Find the range on the stated closed interval.
f(x)=x2,x∈[−2,1]
The minimum is 0 at x = 0. Endpoint values are f(−2) = 4 and f(1) = 1, so the maximum is 4.
For any 0 ≤ y ≤ 4, input −√y lies in [−2, 0] and gives output y. The range is [0, 4].
Transfer and exit task
Find the domain of √(x + 2)/(x − 1). Then find the range of x² when the domain is {−2, 0, 1}; explain why it is not an interval.
Find the domain of √(x + 1)/(x − 2). Explain whether (x² − 1)/(x − 1) and x + 1 are the same function on their natural real domains.
The domain is [−1, 2) ∪ (2, ∞). The two functions are different: the original fraction excludes 1, while x + 1 allows it.
Combines both restrictions with intersection and includes −1.
Retains the excluded input after cancellation.
Questions
Does an input need to pass either gate, or both gates?
Could a simplified expression answer a question the original expression cannot?
Expected responses
Students may use union between restrictions. Ask them to evaluate x = 2 before accepting it.
Endpoint-only checks miss the minimum of x² at 0.
Confusions
Do not turn a finite range {0, 1, 4} into [0, 4].
Bounds alone are insufficient for a complete range; also justify attainability.
Board plan
All restrictions simultaneously → intersection.
Original exclusions survive simplification.
Range: bounds + attainment of every stated output.
