Inverse functions
Problem 13.372 · medium
Find the inverse of \( \displaystyle f(x) = \frac{- 3 x - 1}{- x - 1} \).
- Write y = f(x) and solve for x. Clear the denominator, collect the x terms, and factor x out. Then rename y as x.
- \[ \left. \frac{- 3 x - 1}{- x - 1} \right|_{\substack{ x=\frac{1 - x}{x - 3} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. \frac{1 - x}{x - 3} \right|_{\substack{ x=\left. \frac{- 3 x - 1}{- x - 1} \right|_{\substack{ x=1 }} }} = 1 \]And f⁻¹(f(1)) = 1.✓ Proved
Answer \( f^{-1}(x) = \frac{1 - x}{x - 3} \)
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: passqwen3.6:27b-mlx: fail (misleading) — The solution claims to solve for x but provides no algebraic steps, only a verification that the proposed inverse works. This fails to demonstrate the derivation process requested by 'solve for x'.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The solution claims to solve for x but provides no algebraic steps, only a verification that the proposed inverse works. This fails to demonstrate the derivation process requested by 'solve for x'.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the method for finding the inverse and verifies the result using composition. The algebraic steps described in the sentence are standard and correct.gpt-oss:20b: fail (misleading) 2026-10-17 — The solution claims to prove that f⁻¹ is (1‑x)/(x‑3) by checking only f(f⁻¹(x))=x and a single point f⁻¹(f(1))=1. This does not establish that the proposed function is the inverse for all x; a full algebraic derivation or a check of both compositions for a general x is required.qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The solution provides no derivation for the inverse function, merely stating the method in step 1 and then verifying the result in steps 2 and 3. Verification does not constitute a solution to 'Find the inverse', as it assumes the answer is known. Furthermore, step 2 claims f(f⁻¹(x)) = x generally, but the CAS check shown is likely just a symbolic simplification which might hide domain issues or be insufficient proof without showing the algebraic steps requested in step 1.gpt-oss:20b: pass 2026-10-17
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 2026-10-19 with SymPy 1.14.0.