Inverse functions
Problem 13.376 · medium
Find the inverse of \( \displaystyle f(x) = \ln{\left(x - 4 \right)} + 3 \).
- Write y = f(x) and solve for x. Isolate the log, then exponentiate. Then rename y as x.
- \[ \left. \ln{\left(x - 4 \right)} + 3 \right|_{\substack{ x=e^{x - 3} + 4 }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. e^{x - 3} + 4 \right|_{\substack{ x=\left. \ln{\left(x - 4 \right)} + 3 \right|_{\substack{ x=6 }} }} = 6 \]And f⁻¹(f(6)) = 6.✓ Proved
Answer \( f^{-1}(x) = e^{x - 3} + 4 \)
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 solution verifies the inverse only at a single point (x=6) rather than for all x, which does not establish that f^{-1} is truly the inverse of f. The general proof requires showing f(f^{-1}(x))=x and f^{-1}(f(x))=x for all x in the appropriate domains, not just one example.qwen3.6:27b-mlx: pass — The solution correctly outlines the standard algebraic procedure for finding the inverse and verifies the result using composition. The steps are logically sound and the final answer is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly outlines the standard algebraic procedure for finding the inverse and verifies the result using composition. The steps are logically sound and the final answer is correct.gpt-oss:20b: fail (misleading) 2026-10-19 — The solution verifies the inverse only at a single point (x=6) rather than for all x, which does not establish that f^{-1} is truly the inverse of f. The general proof requires showing f(f^{-1}(x))=x and f^{-1}(f(x))=x for all x in the appropriate domains, not just one example.qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution verifies the result by checking specific points (composition at x and x=6) rather than deriving the inverse function. This teaches that verification is sufficient for finding an inverse, which is false; one must solve for x to determine the function's form.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly outlines the standard algebraic procedure for finding the inverse of a logarithmic function. The verification steps confirm the result is correct.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.