√(x²) = x is true for every x at or above zero, which is why nobody corrects it for years. The figure puts the two graphs on top of each other so the agreement and the disagreement are both visible at once.
Start the slider on the right, at about x = 3, and ask whether √(x²) and x are the same function. They will say yes, and on the evidence in front of them they are right: the V is sitting exactly on the thick line.
Then drag it left through zero. The V turns and the line does not. At x = −3 the readouts are 3 and −3, six apart, and the verdict changes from "the same" to "different".
The sentence to collect is the one at the bottom of the figure: it is right on half the line. That is a better explanation of why this error is so persistent than anything about definitions, and it generalises: students should be suspicious of any rule they have only ever tested on positive numbers.
| Question | Answer |
|---|---|
| 1. √((−3)²) | 3. |
| 2. f(−4) for x³ | −64, which is −f(4), so odd. |
| 3. g(−2) for x² + x | 2, against g(2) = 6, so neither. |
| 4. h(3) for (x + 1)/(x − 1) | 2, and h(2) = 3. |
| 5. Why x² needs a restriction | B. Two inputs share an output. |
Question 4 is chosen so the pair is 3 and 2 rather than 4 and 5/3. Integers make the swap obvious, and a student who has not seen self-inverse before will often notice it without being told, which is worth more than being shown.
1 markThe test stated: f(−x) against f(x) or −f(x).
1 markThe algebra of f(−x), simplified.
1 markThe conclusion, naming odd, even or neither.
On an inverse question, the restricted domain is part of the answer and carries its own mark. "f⁻¹(x) = √x for x ≥ 0" earns it; the formula alone does not. Say this before the exercise, not after it.
| They wrote | What happened |
|---|---|
| −3 on question 1 | The error this page exists for. They have treated rooting as the undo of squaring. Drag the slider rather than explaining. |
| 9 on question 1 | Stopped after squaring. Mechanical, but worth noting whether it recurs: it often means they are reading the symbols left to right rather than innermost first. |
| 64 on question 2 | Lost the sign on an odd power. Then they will conclude x³ is even, so the error cascades into the classification. |
| −12 | Multiplied by 3 instead of cubing. Pure slip. |
| 6 on question 3 | Gave g(2). Either misread, or assumed that since the leading term is even the function is even. |
| −2 on question 3 | Made (−2)² negative. The bracket is the whole issue and it is worth writing it in every time. |
| "Odd" for x² + x | The second common error. They have reasoned from the presence of an odd power rather than testing. The test is three numbers and it settles it. |
| 0.5 on question 4 | Fraction upside down. Check against the asymptotes: h(3) should be near 1 for large x, not near 0. |
| An inverse with no domain | The algebra is right and the answer is incomplete. Give the method marks and withhold the last one, which is what a paper will do. |
"Why is √9 not ±3?" Because √ has to be a function and a function returns one value. The equation x² = 9 has two solutions, and we write x = ±√9 precisely because the √ on its own gives only the positive one. The ± in the quadratic formula exists for the same reason, which is a connection worth making: it is not decoration, it is restoring the root the symbol discarded.
"Can a function be both odd and even?" Yes, exactly one: f(x) = 0. It satisfies both tests trivially. Worth thirty seconds because it stops students treating the two as opposites, which is the reason they resist answering "neither".
"How do I find the domain to restrict to?" Cut at the turning point. For x² the vertex is at 0, so x ≥ 0 or x ≤ 0 both work and either is a valid answer if stated. For (x − 3)² the cut is at 3. Students look for a rule and the rule is the vertex, which they can already find from 2.6.
"Is every function with matching asymptotes self-inverse?" For this family, yes, and the fact is nicer than the hedge. For (ax + b)/(cx + d) the vertical asymptote is x = −d/c and the horizontal is y = a/c, so matching means d = −a; and working out f(f(x)) shows it is x under exactly that same condition. So the two properties are one property. (x + 2)/(x − 1), (2x + 3)/(x − 2) and (3x + 5)/(2x − 3) all have matching asymptotes and all are self-inverse, which is worth setting as a short investigation rather than telling them. 1/x fits too, with both asymptotes at 0.
Do not generalise it past this family. The equivalence is a fact about (ax + b)/(cx + d) and not about rational functions in general, and the test that always works is still f(f(x)) = x.
| Demonstrate | Type √((-3) x² ) and get 3. The machine agrees with the definition and not with the class. That is a cheap and effective thirty seconds, because it removes the suspicion that this is a convention invented by the textbook. |
|---|---|
| Where they stick | Table mode is better than single evaluations for odd and even. Reading f(−2) and f(2) off one screen makes the symmetry visible down the column; evaluating them separately makes it a memory test. |
| The check | For self-inverse, nest it: g(g(4)) returns 4. Try three different inputs. Then say plainly that three values is not a proof, which is the Paper 1 point. |
The square-window setting matters for the y = x symmetry. Without it the reflection looks like a shear and students doubt the claim.
| Step | What |
|---|---|
| 1 | Ask for √((−3)²) cold, on paper, before any theory. Collect both answers. |
| 2 | The figure. Slider right, then left through zero. |
| 3 | √(x²) = |x| as the conclusion, and √ as the non-negative root by definition. |
| 4 | Connect it to 2.5: this is why x² has no inverse. Same fact, two languages. |
| 5 | Restrict the domain and get √x. Insist on writing the restriction. |
| 6 | Odd and even as tests, done on numbers first and algebra second. |
| 7 | x² + x, so "neither" gets said out loud early. |
| 8 | Self-inverse: h(3) = 2 and h(2) = 3, then the y = x symmetry and the matching asymptotes. |
Do not say "rooting undoes squaring". It is the sentence that produces −3, and it is wrong in exactly the way that is hardest to detect, because it works on every positive number anyone tests it on. If you want a short version: squaring loses the sign, and nothing can give it back.
Do not say "odd powers make odd functions". It is true for a single term and false in general, and it is the direct cause of x² + x being called odd. cos x and sin x settle the argument with no powers in sight, so they are worth having ready.