By Christina Sloane – Updated August 30, 2022
The domain of a rational expression is the set of all real numbers that can serve as the independent variable without causing undefined behavior. By applying fundamental algebraic rules and recognizing key restrictions—such as division by zero and non‑real square roots—you can identify the domain for any fraction.
Any expression in the denominator must never equal zero, because division by zero is undefined. For example, in the simple fraction 1/x, the domain is all real numbers except 0.
When a square root appears in the expression, the radicand (the quantity under the square root) must be non‑negative to keep the result real. For (sqrt x)/2, the radicand x ≥ 0, so the domain is all real numbers greater than or equal to 0.
For expressions where the denominator or radicand involves a polynomial, set up an equation to find the values that would violate the rules.
Example 1:
Domain of 1/(x² – 1)
Set the denominator to zero: x² – 1 = 0 → x² = 1 → x = ±1. These values are excluded, so the domain is all real numbers except 1 and –1.
Example 2:
Domain of (sqrt(x – 2))/2
Ensure the radicand is non‑negative: x – 2 ≥ 0 → x ≥ 2. The domain is all real numbers greater than or equal to 2.
Example 3:
Domain of 2/(sqrt(x – 2))
Two restrictions apply: the radicand must be positive (since it’s in the denominator) and the square root itself cannot be zero. Solve:
Radicand positive: x – 2 > 0 → x > 2
\Denominator not zero: sqrt(x – 2) ≠ 0 → x ≠ 2
Both conditions together give the domain: all real numbers greater than 2.
