Among the four options, only g(x) = 2x³ is one-to-one. Squaring, f(x) = x² − 1, maps x and −x to the same output. The table becomes one-to-one only if no two rows share a y-value, since repeated outputs there break the rule. And h fails outright because both (1,4) and (3,4) resolve to y = 4. Each case traces back to a single test: does every output come from exactly one input?
The three tests that decide one-to-one
A function is one-to-one, or injective, when no two distinct inputs share an output. In symbols, if f(a) = f(b) then a = b must follow. Equivalently, every horizontal line crosses the graph at most once. The horizontal line test is the graphical face of the algebra.
Start with option a. Squares kill injectivity on the reals because the squaring operation forgets sign. Both x = 2 and x = −2 produce y = 3, giving two points, (2,3) and (−2,3), on a single horizontal line. The domain matters here. Restrict f to x ≥ 0 or x ≤ 0 and the function becomes one-to-one on that half-line, a repair that shows injectivity is a property of the function-plus-domain pair.
Now the cube. For g(x) = 2x³, set g(a) = g(b). Then 2a³ = 2b³, so a³ = b³, so a = b after taking real cube roots. The cube root inverts the cube across all real numbers, negative inputs included. That is the deep reason the cube passes and the square fails: the cube root is a genuine function on ℝ, while ±√ is two-valued.
For option d, read the pairs as coordinates. The inputs 1, 2, 3, 5 are distinct, so the inputs look clean. Outputs are 4, 7, 4, 9, and the collision sits at y = 4. Each input does satisfy functionality, but injectivity demands the reverse: output-to-input must be unambiguous too. Here y = 4 maps back to both x = 1 and x = 3.
Option b needs its table supplied. Use the column test rather than eyeballing: any repeated value in the output column means not one-to-one. Two identical y entries, say 4 and 4, produce the same collision h has. One structural caveat matters: a finite table proves the property only over listed inputs. Extrapolating to unlisted inputs needs the rule or a shape that supports it.
Why the cube passes and the square fails
The mechanism is invertibility, not graph shape. Every function sends each input to exactly one output. A one-to-one function also sends each output back to exactly one input, so it is invertible over its range. The cube is one-to-one because the equation x³ = c has exactly one real solution for every real c. The square is not, because x² = c has two solutions whenever c > 0.
Monotonicity gives a general checkpoint. A strictly increasing function on an interval is one-to-one there, which explains why g(x) = 2x³ passes on all of ℝ. Strictly decreasing functions qualify the same way. Squares fail monotonicity on all of ℝ: the parabola descends to the vertex and then climbs, so the horizontal line test catches it twice.
The counterintuitive part is that domain, not formula, usually decides the verdict. g(x) = x² on {0, 1, 2, …, 10} is perfectly one-to-one, assigning 0, 1, 4, …, 100 to distinct inputs. f(x) = x² on ℝ is not. A 900-word explainer saying “squaring always fails” is wrong for any finite or nonnegative domain, and that error is common.
Quantified context and boundaries
Precalculus and algebra courses typically cover injectivity in the first two weeks of function-unit instruction. The test itself costs nothing formally: pick any two inputs, set outputs equal, and solve. In practical terms, a repeated y-value in a table is a hard failure for injectivity, while a single instance is enough to disprove the claim.
Domain and range usually follow a baseline in US courses: the default domain is all real numbers unless stated. That default is why a is not one-to-one in its standard form. Restricting to a half-line and naming the range is the usual repair. The repair does not change the formula, which makes the “function-plus-domain pair” phrase worth repeating.
Warning flags to check: two equal outputs in a table, horizontal stretches that catch the same height, and any squared term on an unrestricted domain. Do not mistake one-to-one for onto: a function can be injective without covering every output value, and onto can fail independently. The cube is both on ℝ, since 2x³ can reach any real value. A half-restricted square is injective but not onto over ℝ.
A common professional caveat concerns inverse notation. Writing f⁻¹ assumes the function is one-to-one on its stated domain; otherwise the inverse is a multivalued relation, not a function. In quantitative practice, injectivity is the precondition for unique decoding of labels from values. That is why machine-learning pipelines care about collisions in target outputs.
One-to-one versus related function properties
| Property | Definition | Option a: f(x) = x² − 1 | Option c: g(x) = 2x³ | Option d: h = {(1,4), (2,7), (3,4), (5,9)} |
|---|---|---|---|---|
| One-to-one (injective) | Every y has at most one x | No on ℝ; yes on x ≥ 0 | Yes on ℝ | No (y = 4 twice) |
| Onto (surjective) on ℝ | Every y has at least one x | No (range y ≥ −1) | Yes | No (missing y values) |
| Inverse is a function | Unique x back for each y | No on ℝ; yes restricted | Yes | No |
| Horizontal line test | Line meets graph once | Fails twice | Always passes | Fails at y = 4 |
| Monotone on ℝ | Always rising or always falling | No | Yes, rising | Not applicable |
The comparison row that trips students is the bottom one. A graph can look smooth and clean yet fail injectivity through a single horizontal collision. The table labels smoothness as irrelevant and collision-counting as decisive.
Reading the verdict step by step
The table version of the same logic is the column checklist, applied in order.
Confirm every input appears once so the relation is a function. Scan the output column for repeated values. One duplicate ends the analysis. If no duplicate appears, note the limited evidence: injectivity on listed inputs only. If the exercised points lie on a recognizable curve, infer the rule and test the full domain.
Professional encoders apply the same rule. Hashing a record to a bucket is a function, and collisions exist exactly when two records land in one bucket. A one-to-one encoding is sometimes called perfect hash, and most systems run at under 1-knot-perfect roughly 0 percent of the time when buckets are reused. Restricting the domain, such as fixing record size, keeps the encoding injective and predictable. The lesson transfers directly: the property depends on which inputs are admitted, not on the arithmetic alone.
Practical actions to test a function
The core move costs five minutes. For any candidate, set f(a) = f(b) and solve; if the solutions force a = b, injectivity holds. For g(x) = 2x³ the division is immediate, and the real-root step is legitimate. For f(x) = x² − 1 the solutions allow a = −b, so the claim fails on ℝ. Do the same with a table by scanning the y-column for duplicates. With the exercised points in hand, extrapolate only where the rule supports a full-domain argument.
Then check three thresholds before writing the answer you will defend. Verify whether the domain is stated and applied consistently. Verify every output value in the table before declaring a pass. Verify that your inverse expression exists as a function over the full target range, since finite examples prove less than full-domain claims. If any of those three checks fails, lowercase your conclusion and state the restriction.
End with the repair. If a function is not one-to-one, list the minimal domain change, such as x ≥ 0 for the square, or a signature line like “over the stated domain.” That distinguishes a complete answer from a thin verdict and preempts follow-up grading.
Limits of the verdict
One-to-one is a property of a function with its domain, never of a formula alone. The four options answer cleanly here because option a behaves as unrestricted over ℝ, but the table's status always depends on the supplied data, and it appears secondhand in the exercise. Without the table contents, treat it as a conditional pass pending the duplicate scan. Additionally, a one-to-one verdict echoes an exact mapping from input to output; small input changes can collapse the outputs. A steeper rate arrives with larger coefficients, as with the 2 in 2x³, but for typical algebra ranges the collision risk stays statistical rather than guaranteed.
Also distinguish the map from the pairing structure: sets, tables, and coordinate lists are the same object at different scales. The option-d collision, with y = 4 from x = 1 and x = 3, is a warning about losing information under merging. The pair list is sufficient evidence because two distinct inputs share an output. That is all the fail test requires.