Domain of a function
Problem 13.214 · easy
Find the domain of \( \displaystyle f(x) = \frac{1}{\left(x - 4\right) \left(x + 2\right)} \). Give it in interval notation.
- \[ \left. x^{2} - 2 x - 8 \right|_{\substack{ x=-2 }} = 0 \]At x = -2 the denominator is 0.✓ Proved
- \[ \left. x^{2} - 2 x - 8 \right|_{\substack{ x=4 }} = 0 \]At x = 4 the denominator is 0.✓ Proved
- Keep the x where every fraction has a nonzero denominator, every square root a nonnegative radicand and every logarithm a positive argument.Reviewed
Answer \( (-\infty, -2) \cup (-2, 4) \cup (4, \infty) \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | f evaluated at 321 points in [−10, 10] is real exactly where the claimed domain says |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the points where the denominator is zero and states the general condition for the domain of a rational function. The final answer provided in the problem statement is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the points where the denominator is zero and states the general condition for the domain of a rational function. The final answer provided in the problem statement is correct.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the points where the denominator is zero and states the general principle for finding the domain of a rational function. The logic is sound.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the points where the denominator is zero and states the general rule for finding the domain of a rational function. The logic is sound and leads to the correct answer.gpt-oss:20b: pass 2026-10-15
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/domain_range, checked 2026-10-18 with SymPy 1.14.0.