Higher Level only. The informal version of inverses is SL 2.2.
Ask which order the shop should use, and why. Not which gives the bigger number: which the shop would choose. Somebody will work out that the shop prefers VAT first, and then you have a reason to compute both.
Step the figure across the three prices. The two answers move and the gap does not, and that is the part worth stopping on: the gap is the VAT on the discount, so it cannot depend on the price. Ask what the gap would be if the discount were 200 baht. Fourteen.
The fourth view is the inverse, and the sentence to land is that the steps come off in the reverse order. Socks then shoes going on; shoes then socks coming off. Students who have that will never write the inverse of a composite the wrong way round.
| Question | Answer |
|---|---|
| 1. v(d(500)) | 1.07 × 400 = 428.00. |
| 2. d(v(500)) | 535 − 100 = 435.00. |
| 3. Ticket price behind 963 | 963 ÷ 1.07 + 100 = 1000. |
| 4. g⁻¹(7) on x ≥ 3 | 3 + √9 = 6. |
| 5. Why a restriction is needed | B. Two inputs share an output. |
1 markThe composite written with the inside function applied first.
1 markSimplifying it where asked.
1 markThe inverse, undone in reverse order.
1 markIts domain, or the restriction that makes it exist.
HL questions here are usually built around that last mark and it is a sentence. Train the form: "restricting to x ≥ 3 makes g one-to-one, so the inverse exists and takes the positive root."
| They wrote | What happened |
|---|---|
| 435 on question 1 | Composed in reading order. The error the page is about. |
| 400 | Stopped after the inside function. |
| 535 | Did the VAT and forgot the discount entirely. |
| 393 | Computed 500 − 100 × 1.07, applying the VAT to the discount instead of the price. Order of operations rather than order of functions. |
| 993 | On question 3, added the 100 before dividing. The right steps, the wrong order. |
| 900 | On question 3, stopped at the discounted price. |
| 0 on question 4 | Took the negative root, which is the inverse for the OTHER restriction. Worth praising: it is a correct inverse of a different function. |
| 14 | On question 4, ran g forwards instead of backwards. |
| 5 | On question 4, gave g⁻¹(2) rather than g⁻¹(7). |
"Which one is f and which is g?" In (f ∘ g)(x) the one touching the x runs first. Write f(g(x)) underneath every time until it is automatic; the circle notation is where the confusion lives and the bracket notation is unambiguous.
"Do composites ever commute?" Sometimes, and it is worth one example: two percentage changes do, because multiplication commutes. A flat amount and a percentage do not. So the question is not whether order matters in general but whether these two particular operations interfere.
"Why can't I just swap x and y and solve?" You can, and it is the standard method. The reason this page leads with undoing step by step is that it makes the reversal of ORDER visible, which the swap-and-solve method hides. Teach both and let them check one against the other.
| Stage | What to do |
|---|---|
| Demonstrate | Define d and v once, then evaluate both composites at three different prices in six keystrokes. The gap staying at 7 while the answers move is the demonstration, and it is much faster on the machine than on paper. |
| Where they stick | On the Nspire, := needs ctrl var, and a plain = defines nothing. On the Casio, a composite is written with the Y names from VARS, and typing a literal Y gives a variable rather than a function. |
| The check | Compose a function with its own inverse and confirm you get the input back. One line, and it catches a wrong inverse immediately, which is the only check available in an exam. |
Resist graphing inverses until the window has equal scales. A reflection in y = x looks like a reflection in something else on a default window, and the single visual idea in this sub-topic is lost.
| Step | What |
|---|---|
| 1 | "The shop takes 100 off and adds 7% VAT. Which order should it use?" Let them argue. |
| 2 | Compute both. Step the figure across prices and watch the gap hold. |
| 3 | Introduce the notation only now, with f(g(x)) written under every circle. |
| 4 | The inverse, undone step by step, then the same answer by swap-and-solve. |
| 5 | The check: f then f-inverse returns x. |
| 6 | (x − 3)² − 2, with g(1) and g(5) both 2. The restriction, and which root it chooses. |
Do not read f ∘ g left to right. If you say "f then g" once, it will be in their exam. Say "g first, because it is next to the x", every time.
Do not call the restriction a technicality. It is the mark, and it is also the mathematics: without it the inverse is not a function at all. A student who thinks it is bookkeeping will omit it and lose a quarter of the question.