대수: 정의역, 연립방정식, 부등식 및 함수
도입됨| English | 의미 |
|---|---|
| domain restriction/dəˈmeɪn rɪˈstrɪkʃn/ | 정의역 제한 |
| simultaneous solution/ˌsɪməlˈteɪnɪəs səˈluːʃn/ | 연립해 |
A decision before an answer
- Cancelling x−1 does not make x=1 valid in an expression that originally divided by x−1.
- Your goal: Solve equations while preserving denominator and radical restrictions.
Read the relationship
- Write restrictions before manipulating an equation. A denominator must be nonzero; a real even root needs a nonnegative argument. Cancelling factors preserves the expression only on its original domain. Squaring can introduce candidates because a square loses sign information; verify every candidate in the unsquared relation.
- Analyse inequalities and simultaneous conditions without losing endpoints.
√(x+6)=x has which real solution?
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Squaring gives (x−3)(x+2)=0, but only nonnegative 3 works.
Use the defining rule
- Solving a system requires the same ordered pair to satisfy all relations. Substitution or elimination can expose one, no or infinitely many linear solutions. For an inequality, multiplication or division by a negative reverses direction. For a factored quadratic inequality, order roots and test interval signs, including endpoints only when equality is allowed.
- Evaluate functions and translate verbal relationships into algebra.
−2x<6 is equivalent to:
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Divide by negative −2 and reverse the sign.
Check the conditions
- A function assigns one output to each permitted input. Evaluate the inner function first in a composition, preserving domain restrictions. Read a slope as change in output per input unit and an intercept as the output at input zero only when that input is meaningful. A verbal model should name variables and keep quantities in compatible units.
- Evaluate functions and translate verbal relationships into algebra.
For √(x+2)=x, x≥0. Squaring gives x²−x−2=0, candidates 2 and −1. Only 2 satisfies the original. For (x−1)(x−4)≤0, an upward quadratic is nonpositive on 1≤x≤4, including roots. In the system x+y=10, x−y=2, adding gives 2x=12; x=6 and y=4. For f(t)=2t+1 and g(x)=x², f(g(3))=2·9+1=19, applying g first.
If f(x)=3x−2, f(4)=____.
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Substitute into the complete function rule.
Apply the task format
- For Quantitative Comparison, preserve all allowed values. A variable with x²=4 could be −2 or 2 unless constrained. One counterexample can reject a universal relationship; several favourable substitutions cannot prove it for every real value. Algebraic structure can establish a relationship without exhaustive numerical sampling.
- Evaluate functions and translate verbal relationships into algebra.
A transformed equation supplies candidates, not automatic original solutions. Negative inequality multipliers and strict endpoints need explicit checks.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
분모와 근호에 대한 제한 조건을 유지한 채 방정식 풀이하기
Simplifying (x²−1)/(x−1) removes the original exclusion x=1.
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Cancellation preserves the exclusion from the original denominator.
Keep the distinctions
- domain restriction 定义域限制 — A condition identifying inputs on which the original expression is defined.
- simultaneous solution 联立解 — Values satisfying every relation in a system at the same time.
- Solve equations while preserving denominator and radical restrictions.
- Analyse inequalities and simultaneous conditions without losing endpoints.
- Evaluate functions and translate verbal relationships into algebra.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.