Inverse functions
Problem 13.32 · easy
Find the inverse of \( \displaystyle f(x) = 2 - 4 x \).
- Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.Reviewed
- \[ \left. 2 - 4 x \right|_{\substack{ x=- \frac{x}{4} + 1 \cdot \frac{1}{2} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. - \frac{x}{4} + 1 \cdot \frac{1}{2} \right|_{\substack{ x=\left. 2 - 4 x \right|_{\substack{ x=1 }} }} = 1 \]And f⁻¹(f(1)) = 1.✓ Proved
Answer \( f^{-1}(x) = \frac{1}{2} - \frac{x}{4} \)
✓ 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 identifies the method for finding an inverse function and provides a valid candidate. The verification steps, while limited to specific points rather than a general proof, are sufficient to confirm the result in this context and do not teach false concepts.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the method for finding an inverse function and provides a valid candidate. The verification steps, while limited to specific points rather than a general proof, are sufficient to confirm the result in this context and do not teach false concepts.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the method for finding an inverse function and provides a valid algebraic result. The verification steps, while specific, confirm the correctness of the derived inverse.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the method for finding an inverse function and provides valid verification steps. The stated 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.