How to Find the Domain of a Fraction

Determine impossible values to find the domain of a function.
••• Comstock Images/Comstock/Getty Images

The domain of a fraction refers to all real numbers that the independent variable in the fraction can be. Knowing certain mathematical truths about real numbers and solving some simple algebra equations can help you find the domain of any rational expression.

    Look at the fraction’s denominator. The denominator is the bottom number in the fraction. Since it is impossible to divide by zero, the denominator of a fraction cannot equal zero. Therefore, for the fraction 1/x, the domain is “all numbers not equal to zero,” since the denominator cannot equal zero.

    Look for square roots anywhere in the problem, for example (sqrt x)/2. Since square roots of negative numbers are not real, the values under the square root symbol must be greater than or equal to zero. In our example problem, the domain is “all numbers greater than or equal to zero.”

    Set up an algebra problem to isolate the variable in more complicated fractions.

    For example: To find the domain of 1/(x^2 -1), set up an algebra problem to find the values of x that would cause the denominator to equal 0. X^2-1 = 0 X^2 = 1 Sqrt (x^2) = Sqrt 1 X = 1 or -1. The domain is “all numbers not equal to 1 or -1."

    To find the domain of (sqrt (x-2))/2, set up an algebra problem to find the values of x that would cause the value under the square root symbol to be less than 0. x-2 < 0 x < 2 The domain is “all numbers greater than or equal to 2."

    To find the domain of 2/(sqrt (x-2)), set up an algebra problem to find the values of x that would cause the value under the square root symbol to be less than 0 and the values of x that would cause the denominator to equal 0.

    x-2 < 0 x-2 <0 x < 2

    and

    Sqrt (x-2) = 0 (sqrt (x-2))^2 = 0^2 x-2 = 0 x = 2

    The domain is “all numbers greater than 2.”