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Domain of a Function

CSCA Domain of a Function study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Domain of a Function

1. Concept

The **domain** of a function is the set of all possible values for the independent variable $x$. It is one of the three essential elements of a function.

Simply put, the domain consists of all real numbers $x$ for which the function $y=f(x)$ is **defined** (makes mathematical sense). Finding the domain is essentially solving a system of inequalities.

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2. The "Big 5" Rules for Finding the Domain

For functions defined by analytical expressions, we must adhere to the following operational rules:

| Type | Expression | Condition | Note |

| :--- | :--- | :--- | :--- |

| **Fraction** | $\frac{A(x)}{B(x)}$ | $B(x) \neq 0$ | Denominator $\neq 0$ |

| **Even Root** | $\sqrt[2n]{A(x)}$ | $A(x) \ge 0$ | Radicand $\ge 0$ |

| **Logarithm** | $\log_a A(x)$ | $A(x) > 0$ | Argument $> 0$ |

| **Zero Exponent** | $[A(x)]^0$ | $A(x) \neq 0$ | Base $\neq 0$ |

| **Tangent** | $\tan A(x)$ | $A(x) \neq \frac{\pi}{2} + k\pi$ | $k \in \mathbb{Z}$ |

**Key Principle**: If a function contains multiple restrictions, the final domain is the **intersection** of the solution sets for each condition.

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3. Typical Examples

**Example 1: Combination of Root and Fraction**

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

**Solution**:

Two conditions must be met simultaneously:

1. **Even Root**: $x+1 \ge 0 \Rightarrow x \ge -1$

2. **Denominator**: $x-2 \neq 0 \Rightarrow x \neq 2$

Take the **intersection** of both sets:

**Result**: The domain is $[-1, 2) \cup (2, +\infty)$.

**Example 2: Combination of Logarithm and Root**

Find the domain of $f(x) = \log_2(x-3) + \sqrt{4-x}$.

**Solution**:

Two conditions must be met simultaneously:

1. **Logarithm**: $x-3 > 0 \Rightarrow x > 3$

2. **Even Root**: $4-x \ge 0 \Rightarrow x \le 4$

Take the **intersection** of both sets:

**Result**: The domain is $(3, 4]$.

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4. Common Pitfalls

* **Interval Endpoints**: Pay attention to open intervals `(` vs. closed intervals `[`. A solid dot means included ($\ge, \le$), while a hollow circle means excluded ($>, <, \neq$).

* **Missing Exclusions**: In Example 1, a common mistake is writing $[-1, +\infty)$ and forgetting to exclude $x=2$.

* **Real-world Context**: If the function models a real problem (e.g., number of people, time), ensure $x$ makes practical sense (e.g., positive integers).