Inverse functions
Problem 13.44 · medium
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 1 \right)} + 1 \).
- Write y = f(x) and solve for x. Isolate the log, then exponentiate. Then rename y as x.Reviewed
- \[ \left. \ln{\left(x - 1 \right)} + 1 \right|_{\substack{ x=e^{x - 1} + 1 }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. e^{x - 1} + 1 \right|_{\substack{ x=\left. \ln{\left(x - 1 \right)} + 1 \right|_{\substack{ x=4 }} }} = 4 \]And f⁻¹(f(4)) = 4.✓ Proved
Answer \( f^{-1}(x) = e^{x - 1} + 1 \)
Lines: 2 proved, 1 reviewed. 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 | 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 outlines the standard procedure for finding an inverse function and provides valid verification steps.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly outlines the standard procedure for finding an inverse function and provides valid verification steps.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the method for finding the inverse and provides valid verification steps. The logic is sound.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the method for finding the inverse function and provides a valid verification step. The stated answer is correct.gpt-oss:20b: fail (error) 2026-10-14 — The solution never actually solves for x to find the inverse; it merely plugs the proposed inverse into f and checks that the composition equals x. The step “Subs(log(x - 1) + 1, x, exp(x - 1) + 1) = x” does not demonstrate that exp(x‑1)+1 is the inverse, so the reasoning is incomplete and incorrect.
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.