Inverse functions
Problem 13.39 · medium
Find the inverse of \( \displaystyle f(x) = e^{- 2 x} \).
- Write y = f(x) and solve for x. Isolate the exponential, then take the natural log. Then rename y as x.Reviewed
- \[ \left. e^{- 2 x} \right|_{\substack{ x=\frac{\left(-1\right) \ln{\left(x \right)}}{2} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. \frac{\left(-1\right) \ln{\left(x \right)}}{2} \right|_{\substack{ x=\left. e^{- 2 x} \right|_{\substack{ x=2 }} }} = 2 \]And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = - \frac{\ln{\left(x \right)}}{2} \)
✓ 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 outlines the standard method for finding an inverse function 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 2026-10-17 — The solution correctly outlines the standard method for finding an inverse function and verifies the result using composition. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly outlines the standard method for finding an inverse function and provides valid verification steps.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution omits the crucial domain restriction x > 0 for the inverse function, which is required because the range of f(x) = e^{-2x} is (0, ∞). Without this, the inverse is not well-defined for all real x.gpt-oss:20b: pass 2026-10-14
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.