To find the domain of a function, one needs to determine the set of all possible input values (usually xxx) for which the function is defined without any issues such as division by zero or taking the square root of a negative number. The process involves checking the function's expression and applying specific rules based on the type of function:
- Polynomial functions (linear, quadratic, cubic, etc.) have a domain of all real numbers.
- Square root functions require the expression inside the root to be greater than or equal to zero.
- Rational functions require the denominator to be non-zero.
- Logarithmic functions require the argument to be greater than zero.
The steps to find the domain are generally:
- Check if all real numbers can be inputs.
- Identify and exclude values that cause division by zero or negative values under even roots.
- Account for any other given restrictions on the input values.
- Use inequalities to solve for allowable xxx values, then express the domain typically in interval notation.
For example, for f(x)=1x2−1f(x)=\frac{1}{x^2-1}f(x)=x2−11, the domain excludes x=±1x=\pm 1x=±1 since these make the denominator zero. Thus, its domain is all real numbers except −1-1−1 and 111.
