Inverse functions
Problem 13.380 · easy
Find the inverse of \( \displaystyle f(x) = 5 x - 1 \).
- Write y = f(x) and solve for x. Subtract the constant and divide. Then rename y as x.
- \[ \left. 5 x - 1 \right|_{\substack{ x=\frac{x}{5} + 1 \cdot \frac{1}{5} }} = x \]f(f⁻¹(x)) = x, so this is the inverse.✓ Proved
- \[ \left. \frac{x}{5} + 1 \cdot \frac{1}{5} \right|_{\substack{ x=\left. 5 x - 1 \right|_{\substack{ x=2 }} }} = 2 \]And f⁻¹(f(2)) = 2.✓ Proved
Answer \( f^{-1}(x) = \frac{x}{5} + \frac{1}{5} \)
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 claims that verifying f(f⁻¹(x)) = x is sufficient to prove a function is the inverse, which is false for non-injective functions (e.g., f(x)=x²). A correct proof requires showing both compositions equal x or establishing injectivity.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The solution claims that verifying f(f⁻¹(x)) = x is sufficient to prove a function is the inverse, which is false for non-injective functions (e.g., f(x)=x²). A correct proof requires showing both compositions equal x or establishing injectivity.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the method for finding the inverse of a linear 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 2026-10-17 — The solution correctly describes the standard procedure for finding an inverse function and provides valid verification steps.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.