To find the domain of a real-valued function, you must identify all real numbers for which the function produces a real number output and is mathematically defined. The domain of a function is the complete set of all possible input values (often represented by x) for which the function yields a valid output. A real-valued function is specifically one where for every input in its domain, the output value is a real number (not an imaginary or complex number).
This means you need to identify and exclude any input values that would cause the function to be undefined or produce a non-real number.
Understanding Restrictions for Real-Valued Functions
When determining the domain of a real-valued function, certain mathematical operations impose restrictions on the input values to ensure the output remains a real number. The primary restrictions to consider are:
- Division by Zero: The denominator of a fraction cannot be equal to zero, as division by zero is undefined.
- Even Roots of Negative Numbers: The expression inside an even root (like a square root, fourth root, etc.) must be greater than or equal to zero, because the even root of a negative number is not a real number. For example, for a function like $f(x) = \sqrt{2 - x}$, the term $(2 - x)$ must be set to $\ge 0$ to find the valid domain.
- Logarithms of Non-Positive Numbers: The argument (the input) of any logarithm (natural logarithm, common logarithm, etc.) must be strictly greater than zero. You cannot take the logarithm of zero or a negative number.
Step-by-Step Approach to Finding the Domain
Follow these general steps to determine the domain of a real-valued function:
- Assume All Real Numbers: Begin by assuming the domain is all real numbers, $(-\infty, \infty)$, unless a specific restriction is present.
- Identify Potential Restrictions: Look for the three common scenarios mentioned above:
- Variables in the denominator of a fraction.
- Variables under an even root (square root, fourth root, etc.).
- Variables within the argument of a logarithm.
- Set Up Inequalities or Equations:
- For denominators: Set the denominator not equal to zero and solve.
- For even roots: Set the expression under the radical greater than or equal to zero ($\ge 0$) and solve.
- For logarithms: Set the argument of the logarithm strictly greater than zero ($> 0$) and solve.
- Solve and Exclude: Solve the resulting equations or inequalities. The solutions represent the values that must be excluded from the domain or the range of values that are allowed.
- Express the Domain: Write the final domain using interval notation or set-builder notation. If multiple restrictions exist, the domain is the intersection of all allowed intervals.
Domain Rules for Common Function Types with Examples
Here's a breakdown of how to find the domain for various types of real-valued functions:
1. Polynomial Functions
- Rule: Polynomials, which consist of terms with non-negative integer exponents, have no restrictions on their input values.
- Domain: All real numbers.
- Example: $f(x) = 3x^2 - 5x + 7$
- There are no denominators, even roots, or logarithms.
- Domain: $(-\infty, \infty)$
2. Rational Functions
- Rule: A rational function is a fraction where the numerator and denominator are polynomials. The only restriction is that the denominator cannot be zero.
- Example: $g(x) = \frac{x+2}{x-4}$
- Set the denominator to not equal zero: $x - 4 \neq 0$
- Solve for $x$: $x \neq 4$
- Domain: All real numbers except 4, or $(-\infty, 4) \cup (4, \infty)$.
- For more on rational function domains, see Khan Academy.
3. Radical Functions
- Even Roots (e.g., Square Root, Fourth Root)
- Rule: The expression under an even root must be greater than or equal to zero.
- Example: $h(x) = \sqrt{3x - 6}$
- Set the expression under the radical $\ge 0$: $3x - 6 \ge 0$
- Solve for $x$: $3x \ge 6 \implies x \ge 2$
- Domain: $[2, \infty)$
- Odd Roots (e.g., Cube Root, Fifth Root)
- Rule: Odd roots can take any real number as an input, including negative numbers. There are no restrictions.
- Domain: All real numbers.
- Example: $k(x) = \sqrt[3]{x+1}$
- No restrictions apply.
- Domain: $(-\infty, \infty)$
4. Logarithmic Functions
- Rule: The argument (the expression inside) of a logarithm must be strictly greater than zero.
- Example: $m(x) = \ln(x - 5)$
- Set the argument $> 0$: $x - 5 > 0$
- Solve for $x$: $x > 5$
- Domain: $(5, \infty)$
- For a comprehensive guide on finding domains, including logarithms, refer to Paul's Online Math Notes.
Summary of Domain Rules
For a quick reference, the table below summarizes the rules for finding the domain of common function types:
| Function Type | Restriction Rule | Example Domain |
|---|---|---|
| Polynomial | No restrictions | $(-\infty, \infty)$ |
| Rational | Denominator $\neq 0$ | $f(x)=\frac{1}{x} \implies (-\infty, 0) \cup (0, \infty)$ |
| Even Root Radical | Expression under radical $\ge 0$ | $f(x)=\sqrt{x} \implies [0, \infty)$ |
| Odd Root Radical | No restrictions | $(-\infty, \infty)$ |
| Logarithmic | Argument of log $> 0$ | $f(x)=\ln(x) \implies (0, \infty)$ |
By systematically applying these rules and analyzing the structure of the function, you can accurately determine its domain for real-valued outputs.
