Inverse functions
Problem 13.221 · medium
Find the inverse of \( \displaystyle f(x) = \frac{- 4 x - 2}{- 3 x - 4} \).
- Write y = f(x) and solve for x. Clear the denominator, collect the x terms, and factor x out. Then rename y as x.Reviewed
- \[ \left. \frac{- 4 x - 2}{- 3 x - 4} \right|_{\substack{ x=\frac{2 \left(1 - 2 x\right)}{3 x - 4} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. \frac{2 \left(1 - 2 x\right)}{3 x - 4} \right|_{\substack{ x=\left. \frac{- 4 x - 2}{- 3 x - 4} \right|_{\substack{ x=2 }} }} = 2 \]And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{2 \left(1 - 2 x\right)}{3 x - 4} \)
✓ 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly describes the standard method for finding an inverse function and provides valid verification steps.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly describes the standard method for finding an inverse function and provides valid verification steps.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly describes the standard method for finding an inverse function and verifies the result with composition checks.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly describes the standard method for finding an inverse function and verifies the result with composition checks.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/inverse_function, checked 2026-10-18 with SymPy 1.14.0.