Inverse functions
Problem 13.31 · medium
Find the inverse of \( \displaystyle f(x) = e^{- 4 x} \).
- Write y = f(x) and solve for x. Isolate the exponential, then take the natural log. Then rename y as x.
- \[ \left. e^{- 4 x} \right|_{\substack{ x=\frac{\left(-1\right) \ln{\left(x \right)}}{4} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. \frac{\left(-1\right) \ln{\left(x \right)}}{4} \right|_{\substack{ x=\left. e^{- 4 x} \right|_{\substack{ x=2 }} }} = 2 \]And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = - \frac{\ln{\left(x \right)}}{4} \)
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: passqwen3.6:27b-mlx: fail (misleading) — The solution verifies the result by checking specific points and composition, but it fails to explicitly derive the inverse function as requested. A student should see the algebraic steps (y = e^(-4x) => ln(y) = -4x => x = -ln(y)/4) rather than just a verification.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-17 — The solution verifies the result by checking specific points and composition, but it fails to explicitly derive the inverse function as requested. A student should see the algebraic steps (y = e^(-4x) => ln(y) = -4x => x = -ln(y)/4) rather than just a verification.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. The logic is sound and the result is correct.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the method for finding the inverse and verifies the result using composition. The steps are logically sound and the final answer is correct.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.