1. **Identify the Function:** Write down the function you are given or want to analyze. For example, let's say we have the function \(f(x) = \frac{1}{x^2 - 4}\).
2. **Identify Excluded Values:** Look for any values of \(x\) that would make the function undefined. These typically include:
- Division by zero: Values of \(x\) that would make the denominator of a fraction equal to zero.
- Square roots of negative numbers: Values of \(x\) that would result in taking the square root of a negative number (for real numbers).
- Other expressions with restricted domains: For example, natural logarithms or square roots.
3. **Exclude Invalid Input Values:** Exclude any values of \(x\) that would make the function undefined. These values are not part of the domain.
4. **Write the Domain:** Write down the set of all valid input values of \(x\) that make the function defined. This set represents the domain of the function.
Let's apply these steps to the example function \(f(x) = \frac{1}{x^2 - 4}\):
1. **Identify the Function:** \(f(x) = \frac{1}{x^2 - 4}\).
2. **Identify Excluded Values:** In this function, the denominator \(x^2 - 4\) cannot be zero. Therefore, \(x\) cannot be \(2\) or \(-2\), because \(2^2 - 4 = 4 - 4 = 0\) and \((-2)^2 - 4 = 4 - 4 = 0\). Division by zero is undefined.
3. **Exclude Invalid Input Values:** Exclude \(x = 2\) and \(x = -2\) from the domain.
4. **Write the Domain:** The domain of the function \(f(x) = \frac{1}{x^2 - 4}\) is all real numbers except \(2\) and \(-2\), which can be written as:
\[ \text{Domain: } x \in (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \]
This represents all real numbers except \(2\) and \(-2\). This is the domain of the function.