Inverse functions
Problem 13.371 · easy
Find the inverse of \( \displaystyle f(x) = 4 x - 3 \).
- Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.
- \[ \left. 4 x - 3 \right|_{\substack{ x=\frac{x}{4} + \frac{3}{4} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. \frac{x}{4} + \frac{3}{4} \right|_{\substack{ x=\left. 4 x - 3 \right|_{\substack{ x=1 }} }} = 1 \]And f⁻¹(f(1)) = 1.✓ Proved
Answer \( f^{-1}(x) = \frac{x}{4} + \frac{3}{4} \)
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: passqwen3.6:27b-mlx: fail (misleading) — The solution verifies the answer by checking specific cases (composition equals identity) rather than deriving the inverse function. This teaches that verification is equivalent to derivation, which is logically incorrect and insufficient for finding an unknown inverse.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The solution verifies the answer by checking specific cases (composition equals identity) rather than deriving the inverse function. This teaches that verification is equivalent to derivation, which is logically incorrect and insufficient for finding an unknown inverse.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly describes the standard method for finding an inverse function and verifies the result using composition. The logic is sound and the steps are clear.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the standard method for finding the inverse of a linear function and verifies the result using composition.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.