KBAT: Modelling with Composite Functions

Q: Does fg(x) mean the same as gf(x) in a modelling problem?

Usually not. fg(x)=f(g(x))fg(x)=f(g(x))fg(x)=f(g(x)) applies ggg first, while gf(x)=g(f(x))gf(x)=g(f(x))gf(x)=g(f(x)) applies fff first. In a real situation that is the difference between discounting before or after a rebate, and here it changes the final price by a fixed RM20. Always write the inner function first and keep the order the story describes.

Q: Can I still score if I set the composite up the wrong way round?

Yes. Add Math Paper 2 is 2 hours 30 minutes and 100 marks with analytic marking, so method marks are given line by line. A correct function rule, a valid substitution and honest working can earn marks even if the order slips, though naming the order first is the surest way to protect them.

What makes this a KBAT question

A routine composite question hands you $f$ and $g$ and says 'find $gf(x)$ '. A modelling KBAT question hides the functions inside a situation, two store offers, two conversion stages, two machines on a line, and asks you to build the composite yourself, then decide something with it.

That is Kemahiran Berfikir Aras Tinggi, higher-order thinking: the composition itself is ordinary Form 4 work, but choosing which process acts first, tracking units, and interpreting the outcome are not spelt out for you. In Add Math this rewards students who see a function as a machine that acts on an input, so that stacking two machines has a real meaning, rather than students who only push symbols around.

The unfamiliar wrapper, not the algebra inside it, is what makes the question hard.

One worked problem, in the style of Paper 2

This is an original question written in the style of SPM Paper 2. Try it yourself before reading the solution.

Q1[7 marks]

An electronics store offers two promotions on a laptop with list price RM $x$ . Promotion A is a flat cash rebate of RM200, modelled by $f(x)=x-200$ .

Promotion B is a 10% member discount, modelled by $g(x)=0.9x$ . A customer may apply both, in either order.

(a) Write the final price as a composite function for each order: Promotion B followed by A, and Promotion A followed by B. (b) A laptop is listed at RM3000.

Find the final price under each order, and state which order the customer should choose. (c) Show that one order is always cheaper than the other by the same fixed amount, whatever the list price, and explain why in terms of the promotions.

Show worked solution

Understand. We have two functions of a ringgit price: $f$ subtracts a fixed rebate, and $g$ multiplies by $0.9$ for a 10% discount.

Both take ringgit to ringgit, so either can act first. We must build both composites, compare them at RM3000, and then compare them in general.

Plan. 'B followed by A' means apply $g$ first, then $f$ , that is $f(g(x))$ , written $fg(x)$ .

'A followed by B' means $g(f(x))$ , written $gf(x)$ . Build both, substitute $x=3000$ , then compute the general difference $gf(x)-fg(x)$ .

Execute and check. (a) B then A applies $g$ first:

fg(x)=f(g(x))=f(0.9x)=0.9x-200

A then B applies $f$ first:

gf(x)=g(f(x))=g(x-200)=0.9(x-200)=0.9x-180

(b) Substitute $x=3000$ into each:

fg(3000)=0.9(3000)-200=2700-200=2500

gf(3000)=0.9(3000)-180=2700-180=2520

So B-then-A gives RM2500 and A-then-B gives RM2520. The customer should take the 10% discount first and the rebate second, paying RM2500, RM20 less.

gf(x)-fg(x)=(0.9x-180)-(0.9x-200)=20

The difference is a constant RM20, independent of $x$ : the $0.9x$ terms cancel. So B-then-A is always RM20 cheaper.

The reason is that a 10% discount applied after the rebate also discounts the rebate itself, and $10\%$ of RM200 is RM20. Applying the discount first protects the full RM200 rebate; applying it last quietly shrinks the rebate by RM20.

Check. Try a different price, $x=1500$ : $fg(1500)=0.9(1500)-200=1150$ and $gf(1500)=1350-180=1170$ .

The difference is again RM20, confirming the result. (The model needs $0.9x\ge 200$ , i.e. $x\ge \text{RM}223$ roughly, so the rebate does not push the price below zero, true for any realistic laptop.)

Finding a sensible first step

When a question buries functions inside a story, the dependable first move is to name each process as a function with its own rule before combining anything. Ask, 'what does this offer do to a price?'

A flat RM200 rebate subtracts, so $f(x)=x-200$ . A 10% discount multiplies by $0.9$ , so $g(x)=0.9x$ .

Write both explicitly. Only then decide the order the question describes, and translate 'B followed by A' into $f(g(x))$ , the inner function acts first.

Keeping units in view protects you here: both offers take ringgit to ringgit, so either order is meaningful, but you must never add ringgit to a count of items or to a percentage. Once the two rules are on paper and the order is fixed, the composite is a single substitution, and every later part rests on solid ground.

What markers reward

Marking is analytic, so method marks are awarded line by line. On a composite-modelling question a marker looks for:

Each process translated into a correct function rule, subtraction for the rebate, $\times 0.9$ for the 10% discount.
The order read correctly,'B then A' written as $f(g(x))$ , with the inner function applied first.
Both composites simplified to a single expression, such as $0.9x-200$ and $0.9x-180$ .
Correct substitution of RM3000 and a clear decision stated, take the discount first.
The general difference computed, $gf(x)-fg(x)=20$ , shown to be independent of $x$ .
An interpretation in words, why applying the percentage last erodes the rebate by RM20.

How a teacher helps

Modelling questions reward students who translate carefully and then explain, and both habits grow fastest with feedback. In a one-to-one lesson our teachers ask you to define each function in words before writing symbols, to say aloud which process acts first, and to check units at every join.

We linger on the interpretation, why the order changes the price by a fixed RM20, because that sentence earns the reasoning mark. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

Message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.

Get 1-to-1 help.

Book a Trial Class

Frequently asked questions

Does

fg(x)

mean the same as

gf(x)

in a modelling problem?

Usually not. $fg(x)=f(g(x))$ applies $g$ first, while $gf(x)=g(f(x))$ applies $f$ first.

In a real situation that is the difference between discounting before or after a rebate, and here it changes the final price by a fixed RM20. Always write the inner function first and keep the order the story describes.

How do I keep track of units when I chain functions?

Check that the output units of the inner function match the input the outer function expects. Both offers here map ringgit to ringgit, so either order is valid.

If one function turned ringgit into a count of items, you could not then subtract ringgit from it, a units mismatch is a common sign the composite is built the wrong way round.

Why does the cheaper order save the same amount at every price?

Because a percentage discount applied after a fixed rebate also discounts the rebate. Ten percent of the RM200 rebate is RM20, so applying the discount last always costs RM20 more, whatever the list price.

The algebra shows it cleanly: $gf(x)-fg(x)=20$ , with the $x$ terms cancelling.

Can I still score if I set the composite up the wrong way round?

Yes. Add Math Paper 2 is 2 hours 30 minutes and 100 marks with analytic marking, so method marks are given line by line.

A correct function rule, a valid substitution and honest working can earn marks even if the order slips, though naming the order first is the surest way to protect them.

Source:SRC-DSKP-EN SRC-FORMAT

Written by the spmaddmath.com.my editorial team.· Last updated 5 September 2026