Identify all numbers for which the function is not defined. For instance, in the problem f(x) = sqrt(x-4), the function is not defined for all values of x less than 4. These values will result in you taking the square root of a negative number. Similarly, for the problem f(x) = 1/(x^2-2), the function is undefined for the x-values sqrt(2) or -sqrt(2), both of which will make the denominator equal zero.
Write the set, in mathematical notation, of all values of x for which the function is defined. The domain of f(x) = sqrt(x-4) is equal to [4, infinity), in which the "[" indicates that the set is inclusive of 4. For the function f(x) = 1/(x^2-2), the domain is equal to (-infinity, -sqrt(2)), (-sqrt(2), +sqrt(2)), (+sqrt(2), infinity), indicating that all values of x less than the negative square root of 2, between the negative and positive square roots of 2, and greater than the square root of 2 are in the function's domain.
Graph the function to check your work. Plug in values of x, and determine for which ones the function is defined. Graph the x and y coordinates and verify that the values of x, forming your graph, are the same as the ones in your domain.