How do you determine the domain of a function? like how did you know it was real numbers

Determining the domain of a function involves identifying all the possible values of the input variable(s) for which the function is well-defined (i.e., the function produces a valid output). Here’s a step-by-step approach:

### 1. **Consider Basic Function Types:**
   - **Polynomial functions** (e.g., \( f(x) = x^2 + 2x + 1 \)): The domain of a polynomial is typically all real numbers \( \mathbb{R} \) because polynomials can take any real number as an input.
   - **Rational functions** (e.g., \( f(x) = \frac{1}{x-2} \)): The domain excludes values that make the denominator zero because division by zero is undefined.
   - **Radical functions** (e.g., \( f(x) = \sqrt{x} \)): The domain includes only values that make the expression inside the square root non-negative (for real-valued functions).
   - **Logarithmic functions** (e.g., \( f(x) = \log(x) \)): The domain includes only positive real numbers since the logarithm of zero or a negative number is undefined.
   - **Trigonometric functions** (e.g., \( f(x) = \tan(x) \)): The domain excludes values where the function is undefined, such as points where the denominator of a trigonometric identity is zero.

### 2. **Check for Division by Zero:**
   Identify any fractions in the function. The domain must exclude values of \( x \) that cause the denominator to be zero. For example, in \( f(x) = \frac{1}{x-2} \), the domain is \( \mathbb{R} \setminus \{2\} \).

### 3. **Check for Square Roots of Negative Numbers:**
   For functions involving square roots, cube roots, or other even roots, ensure the expression inside the root is non-negative when considering real-valued functions. For example, \( f(x) = \sqrt{x-3} \) has a domain of \( x \geq 3 \).

### 4. **Check for Logarithms:**
   Ensure that the argument of a logarithm is positive. For example, \( f(x) = \log(x-1) \) has a domain of \( x > 1 \).

### 5. **Consider Compositions of Functions:**
   If the function is a composition of multiple functions (e.g., \( g(f(x)) \)), the domain is determined by both the inner function and the composite function. Make sure the output of the inner function lies within the domain of the outer function.

### 6. **Use Interval Notation:**
   Once you've identified the valid inputs, express the domain using interval notation. For instance, for \( f(x) = \frac{1}{x-2} \), the domain is \( (-\infty, 2) \cup (2, \infty) \).

### **Applying to the Given Problems:**
- **a.** \((x + 1)^2 = x^2 + 1\): Since this involves only polynomials, which are defined for all real numbers, the domain is \( \mathbb{R} \).
- **b.** \((2x - 1)^2 = 4x^2 - 4x + 1\): Again, this is a polynomial equation, so the domain is \( \mathbb{R} \).

**Summary:** The domain of a function is determined by ensuring the function is defined for the given input. For many elementary functions like polynomials, the domain is often all real numbers, but for others involving fractions, square roots, or logarithms, additional restrictions apply.

Would you like to explore this further or have any more questions?

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Here are 5 related questions you might consider:
1. How do you determine the domain of a function involving both square roots and fractions?
2. What is the domain of the function \( f(x) = \sqrt{x - 2} + \frac{1}{x - 3} \)?
3. How can the concept of domain be extended to complex numbers?
4. What is the relationship between the domain and range of a function?
5. How do you determine the domain of a piecewise function?

**Tip:** When finding the domain, it helps to first look for restrictions in the function (like division by zero or taking the square root of a negative number). This will guide you in identifying the correct domain.

Learn how to determine the domain of a function with a detailed step-by-step guide. This explanation covers various types of functions including polynomial, rational, radical, logarithmic, and trigonometric functions. Suitable for advanced high school students and anyone interested in mastering function domains.

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