定义域dìngyì yù

Core Concept

The domain of a function is the set of all possible values that the independent variable can take. Simply put, it's "what inputs the function can accept."

Mathematical Definition

For function $y = f(x)$, the domain is the set of all $x$ values for which the function is defined, denoted as $D_f$ or $\text{dom}(f)$:

$D_f = \{x \mid f(x) \text{ is defined}\}$

Principles for Finding Domain

1. Rational Functions: Denominator ≠ 0

$f(x) = \frac{1}{x-2}$

Domain: $x - 2 \neq 0$, so $x \neq 2$

$D_f = (-\infty, 2) \cup (2, +\infty)$

2. Even Roots: Radicand ≥ 0

$f(x) = \sqrt{x - 1}$

Domain: $x - 1 \geq 0$, so $x \geq 1$

$D_f = [1, +\infty)$

3. Logarithms: Argument > 0

$f(x) = \log_2(x + 3)$

Domain: $x + 3 > 0$, so $x > -3$

$D_f = (-3, +\infty)$

4. Zero Exponents: Base ≠ 0

$f(x) = (x-1)^0$

Domain: $x - 1 \neq 0$, so $x \neq 1$

5. Real-World Problems: Meaningful Values

Area, length, time must be positive.

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus and Chinese standardized test formats to help students familiarize themselves with question types and problem-solving approaches.

Example 1: Basic (Difficulty ★★☆☆☆)

Find the domain of $f(x) = \sqrt{x+2}$.

Options:

• A. $x > -2$
• B. $x \geq -2$
• C. $x > 0$
• D. $x \geq 0$

Solution:

Even root requires radicand ≥ 0: $x + 2 \geq 0$ $x \geq -2$

Answer: B

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Example 2: Intermediate (Difficulty ★★★☆☆)

Find the domain of $f(x) = \frac{1}{\sqrt{4-x^2}}$.

Solution:

Must satisfy:

1. $4 - x^2 > 0$ (denominator ≠ 0 and radicand > 0)
2. Solve: $x^2 < 4$, so $-2 < x < 2$

Answer: $(-2, 2)$

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Example 3: Advanced (Difficulty ★★★★☆)

Find the domain of $f(x) = \frac{\sqrt{x-1}}{\log_2(3-x)}$.

Solution:

Must satisfy:

1. $x - 1 \geq 0$ → $x \geq 1$
2. $3 - x > 0$ → $x < 3$
3. $\log_2(3-x) \neq 0$ → $x \neq 2$

Combined: $x \in [1, 2) \cup (2, 3)$

Common Mistakes

❌ Mistake 1: $\sqrt{x^2} = x$

Correction: $\sqrt{x^2} = |x|$, not $x$!

❌ Mistake 2: Forgetting denominator ≠ 0

For $f(x) = \frac{x}{x-1}$, must ensure $x \neq 1$.

❌ Mistake 3: Logarithm argument > 0, not ≥ 0

For $f(x) = \ln(x)$, domain is $x > 0$, not $x \geq 0$!

Study Tips

1. ✅ Check systematically: Rational, radical, logarithmic conditions
2. ✅ Find intersection: Multiple conditions → take intersection
3. ✅ Interval notation: Use proper interval notation
4. ✅ Real meaning: Consider practical constraints in word problems

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💡 Exam Tip: Domain is fundamental to function problems. Almost every function question involves it. Master all types!