Welcome to our lesson on function domains. The domain of a function is the complete set of all possible input values, or x-values, for which the function is defined and produces a real number output. Understanding domains is crucial for working with functions correctly.
Not all functions have domains that include all real numbers. Many functions have restrictions on their domains due to mathematical limitations. For example, the function f of x equals one over x cannot have x equal to zero, because division by zero is undefined. This creates a vertical asymptote at x equals zero.
Square root functions have specific domain restrictions. For the function f of x equals square root of x minus 1, we need the expression under the radical to be non-negative. This means x minus 1 must be greater than or equal to zero, so x must be greater than or equal to 1. The domain starts at x equals 1 and extends to positive infinity.
Logarithmic functions also have domain restrictions. For the natural logarithm function f of x equals ln of x, the argument must be positive. This means x must be greater than zero. The function is undefined for zero and all negative numbers, creating a vertical asymptote at x equals zero. The domain includes only positive real numbers.