A shop takes 100 baht off and adds 7% VAT. On a 1,000 baht ticket price that comes to 963 one way round and 970 the other. The seven baht is the VAT on the discount, and it is seven baht at every price.
The same two operations. Only the order changes, and the gap never does.
Call the two operations d for discount and v for VAT:
Then the two orders are two different composite functions:
Subtract them and the x disappears: the gap is 7, whatever the price. That is the 7% VAT on the 100 baht that one order charges and the other does not.
(f ∘ g)(x) means f(g(x)): g happens first. The function written on the inside runs first, which is the opposite of the reading order and the reason people get it backwards. Read it right to left, like putting on socks before shoes.
A customer paid 963 baht. What was the ticket price? Undo the chain, in the opposite order, undoing each step:
As a formula, if y = 1.07(x − 100) then x = y/1.07 + 100. Notice the order reversed: the inverse of a composite undoes the last step first. Socks then shoes going on, shoes then socks coming off.
The defining property is the one worth checking every time:
(f ∘ f⁻¹)(x) = (f⁻¹ ∘ f)(x) = x
If putting a number through both does not give you the number back, one of the two is wrong. It costs one substitution.
Take g(x) = (x − 3)² − 2.
The fix is to cut the domain so each output happens once. The vertex is at x = 3, so restrict to x ≥ 3 (or to x ≤ 3, which gives a different inverse). With x ≥ 3:
Check: g⁻¹(2) = 3 + √4 = 5, and g(5) = 2. And g⁻¹(7) = 3 + √9 = 6.
Choosing the root is the whole restriction. If you restrict to x ≤ 3 then x − 3 is negative, so the inverse is 3 − √(x + 2) and g⁻¹(2) is 1 instead of 5. Both are correct inverses of different functions. A question that restricts the domain for you has told you which sign to take, and students who write the plus out of habit lose the mark on half of them.
Define the two functions once and the machine will compose them either way round, which makes the seven baht a thing you can see rather than a thing you are told.
When you may use it. Applications. A calculator is allowed in every paper.
The mark people lose. Composing in the reading order. (f ∘ g)(x) is f(g(x)), so g runs first, and a student who works left to right computes the other composite and gets a number that is wrong by exactly the amount this page is about. Write the brackets out before touching the machine. The other one is forgetting that the inverse of a composite reverses the order as well as each step: undo the last operation first.
1. With d(x) = x − 100 and v(x) = 1.07x, what is v(d(500))? Give it to 2 decimal places.
2. And d(v(500)), to 2 decimal places?
3. A customer paid 963 baht after the discount and then VAT. What was the ticket price, in baht?
4. For g(x) = (x − 3)² − 2 restricted to x ≥ 3, what is g⁻¹(7)?
5. Why does g(x) = (x − 3)² − 2 need a restricted domain before it has an inverse?
1 markThe composite written correctly, with the inside function applied first.
1 markSimplifying it, where the question asks.
1 markThe inverse, with the steps undone in reverse order.
1 markIts domain, or the restriction that makes it exist.
That last mark is the one HL questions are built around, and it is a sentence. If you restrict a domain, say which half you took and why.
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