Inverse functions
Problem 13.375 · medium
Find the inverse of \( \displaystyle f(x) = \frac{- x - 2}{4 - 2 x} \).
- Write y = f(x) and solve for x. Clear the denominator, collect the x terms, and factor x out. Then rename y as x.
- \[ \left. \frac{- x - 2}{4 - 2 x} \right|_{\substack{ x=\frac{2 \left(2 x + 1\right)}{2 x - 1} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. \frac{2 \left(2 x + 1\right)}{2 x - 1} \right|_{\substack{ x=\left. \frac{- x - 2}{4 - 2 x} \right|_{\substack{ x=1 }} }} = 1 \]And f⁻¹(f(1)) = 1.✓ Proved
Answer \( f^{-1}(x) = \frac{2 \left(2 x + 1\right)}{2 x - 1} \)
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 (error) — The stated answer is incorrect; the correct inverse is (2x+1)/(1-2x). The provided solution claims the wrong answer is correct and supports this false claim with a single numerical check (f(f⁻¹(1))=1) which happens to hold for the wrong answer by coincidence, failing to verify the general identity.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The stated answer is incorrect; the correct inverse is (2x+1)/(1-2x). The provided solution claims the wrong answer is correct and supports this false claim with a single numerical check (f(f⁻¹(1))=1) which happens to hold for the wrong answer by coincidence, failing to verify the general identity.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-17 — The stated answer is incorrect; the correct inverse is (2x+1)/(1-2x). The solution claims the stated answer is correct based on a single point check (x=1), which is insufficient to verify the function and masks the algebraic error.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-17 — The stated answer is incorrect. The correct inverse is f^{-1}(x) = \frac{2x+2}{2x-1}, whereas the solution claims \frac{2(2x+1)}{2x-1}. The verification steps provided in the solution are false (e.g., f(f^{-1}(x)) \neq x for the claimed inverse), indicating a calculation error in the derivation that was not shown.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.