How to Find the Domain of a Function (With Examples)
The domain of a function is the set of inputs that make the function valid. Finding it comes down to three checks: no division by zero, no negative under an even root, no zero or negative log argument.
Quick answer
Start with all real numbers. Exclude values that make denominators zero, even-root expressions negative, or log arguments zero or negative. What’s left is the domain.
Case 1 — Polynomials
Polynomials like f(x) = 3x² + 5x − 2 are defined for every real number. Domain = (−∞, ∞).
Case 2 — Rational functions
For f(x) = 1 / (x − 3), set x − 3 = 0 to find x = 3 is excluded. Domain: x ≠ 3.
Tip: factor the denominator first. x² − 4 hides TWO excluded values: 2 and −2.
Case 3 — Square roots
For f(x) = √(x − 5), require x − 5 ≥ 0 → x ≥ 5. Domain: [5, ∞).
Case 4 — Logarithms
For f(x) = log(x + 2), require x + 2 > 0 → x > −2. Domain: (−2, ∞).
Case 5 — Multiple restrictions
For f(x) = √(x − 1) / (x − 4), require x ≥ 1 AND x ≠ 4. Domain: [1, 4) ∪ (4, ∞).
Common mistakes
- Forgetting odd roots have no restrictions.
- Using > instead of ≥ for square roots.
- Excluding values that aren’t actually excluded.