The domain of a function is the set of all possible input values (usually \( x \)-values) for which the function is defined. To identify the domain of a set of numbers, follow these steps:

### Steps to Identify the Domain

1. **Determine the Type of Function:**
   - Is it a polynomial, rational, radical, exponential, logarithmic, or trigonometric function?

2. **Identify Any Restrictions:**
   - **Polynomials:** The domain is all real numbers, \( \mathbb{R} \), because polynomials are defined for all real numbers.
   - **Rational Functions:** The domain excludes values that make the denominator zero.
   - **Radical Functions:** The domain excludes values that make the expression inside an even root (like a square root) negative.
   - **Exponential Functions:** The domain is all real numbers, \( \mathbb{R} \).
   - **Logarithmic Functions:** The domain includes only positive numbers because the logarithm of a non-positive number is undefined.
   - **Trigonometric Functions:** The domain can vary based on the specific function (e.g., \(\tan(x)\) is undefined at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \)).

3. **Express the Domain:**
   - Use interval notation, set-builder notation, or a verbal description to express the domain.

### Examples

1. **Polynomial Function:**
   - \( f(x) = x^2 + 3x + 2 \)
   - **Domain:** \( \mathbb{R} \) (all real numbers)

2. **Rational Function:**
   - \( g(x) = \frac{1}{x-3} \)
   - **Domain:** \( \mathbb{R} \setminus \{3\} \) (all real numbers except \( x = 3 \))

3. **Radical Function:**
   - \( h(x) = \sqrt{x-5} \)
   - **Domain:** \( x \geq 5 \) or \([5, \infty) \)

4. **Exponential Function:**
   - \( j(x) = 2^x \)
   - **Domain:** \( \mathbb{R} \) (all real numbers)

5. **Logarithmic Function:**
   - \( k(x) = \log(x-2) \)
   - **Domain:** \( x > 2 \) or \((2, \infty) \)

6. **Trigonometric Function:**
   - \( m(x) = \tan(x) \)
   - **Domain:** \( x \neq \frac{\pi}{2} + n\pi \) for any integer \( n \)

### General Steps for Any Function
1. Identify if there are any restrictions (like division by zero, even roots of negative numbers, or logarithms of non-positive numbers).
2. Exclude the values that cause these restrictions from the domain.

Would you like more details on any specific type of function or have any particular function in mind?

Here are 5 questions you might consider next:
1. How can I determine the domain of a composite function?
2. What are some common pitfalls when identifying domains?
3. How do you graph the domain of a function?
4. Can you show an example with a piecewise function?
5. What is the domain of a function with multiple variables?

**Tip:** Always double-check for values that could make a denominator zero or an expression under a square root negative when determining the domain of a function.

Learn how to identify the domain of a function with this step-by-step guide. Understand the domain for polynomial, rational, radical, exponential, logarithmic, and trigonometric functions. Suitable for high school students and anyone interested in understanding function domains.

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