Inverse functions
Problem 13.36 · easy
Find the inverse of \( \displaystyle f(x) = - x - 3 \).
- Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.
- \[ \left. - x - 3 \right|_{\substack{ x=- x - 3 }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. - x - 3 \right|_{\substack{ x=\left. - x - 3 \right|_{\substack{ x=2 }} }} = 2 \]And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = - x - 3 \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | f then the answer, in floating point, returns the starting x |
Reviewers
gpt-oss:20b: fail (misleading) — The first sentence incorrectly describes solving for the inverse: after solving y = -x-3 one obtains x = -y-3, and renaming y as x gives f^{-1}(x)= -x-3, not the erroneous x = -x-3 implied by the wording. This misleads a student about the correct algebraic manipulation.qwen3.6:27b-mlx: fail (error) — The solution claims f(x) = -x - 3 is its own inverse, but the actual inverse is f^{-1}(x) = -x + 3. The verification steps are algebraically false (e.g., f(f(x)) = x - 3, not x), so the 'proved' equations are incorrect.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution claims f(x) = -x - 3 is its own inverse, but the actual inverse is f^{-1}(x) = -x + 3. The verification steps are algebraically false (e.g., f(f(x)) = x - 3, not x), so the 'proved' equations are incorrect.gpt-oss:20b: fail (misleading) 2026-10-17 — The first sentence incorrectly describes solving for the inverse: after solving y = -x-3 one obtains x = -y-3, and renaming y as x gives f^{-1}(x)= -x-3, not the erroneous x = -x-3 implied by the wording. This misleads a student about the correct algebraic manipulation.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims f^{-1}(x) = -x - 3, but the actual inverse is f^{-1}(x) = -x + 3. The verification steps in lines 2 and 3 are algebraically false (e.g., f(f(x)) = x - 6, not x), so the 'proved' tags are incorrect and the conclusion is wrong.gpt-oss:20b: fail (error) 2026-10-14 — The sentence claims to solve for x and rename y as x, but the algebraic step is incorrect: solving y = -x-3 yields x = -y-3, and renaming y to x gives x = -x-3, not the correct inverse -x-3. The setup and reasoning are flawed.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims f(x) is its own inverse without showing the derivation, and the verification steps are presented as equations rather than logical deductions. While f(x) = -x - 3 is indeed its own inverse, the method described ('Subtract the constant and divide') is not explicitly applied to show the result, making the solution incomplete and potentially misleading about the process.gpt-oss:20b: pass 2026-10-14
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/inverse_function, checked 2026-10-17 with SymPy 1.14.0.