A machine plugged into a machine: Picture two machines on a conveyor belt. The number goes into the inner machine g first, and whatever comes out is dropped straight into the outer machine f.
That chain is written (f∘g)(x) = f(g(x)) — read it right-to-left: g first, then f.
The small circle ∘ just means 'composed with'. To evaluate, work from the inside out: compute g(x), then apply f to that result.
IB-style question — evaluate a composite
A factory's profit depends on output through f(x) = 3x − 5, and output depends on staff through g(x) = x² + 1.
Find (f∘g)(2).
Step by step
- Do the inside g first: substitute x = 2 into g.
- Feed that result into the outer function f.
Final answer
(f∘g)(2) = 10 — the result of running 2 through g and then f.
IB-style question — build the formula
With f(x) = 3x − 5 and g(x) = x² + 1, find an expression for (f∘g)(x).
Step by step
- Start from the definition: f∘g means f of g(x).
- Wherever f has its input, put the whole inner expression x² + 1.
- Expand and simplify.
Final answer
(f∘g)(x) = 3x² − 2. Check: at x = 2 this gives 3(4) − 2 = 10, matching above.
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f∘g is usually NOT g∘f: Swapping the order changes the answer, because you change which machine runs first.
Think of a shop sale: one machine takes 20% off (multiply by 0.8) and another adds $5 delivery (add 5). Taking 20% off then adding delivery is not the same total as adding delivery then taking 20% off the bigger amount.
In the exam this shows up as 'which order is cheaper / better?' — you compute both composites and compare, then interpret the result in words.
IB-style question — which order is cheaper?
A chair costs $200. A store offers a discount d(x) = 0.8x (20% off) and a voucher v(x) = x − 30 ($30 off).
Is it cheaper to apply the discount first or the voucher first?
Step by step
- Discount first, then voucher: (v∘d)(200) = v(0.8 × 200).
- Voucher first, then discount: (d∘v)(200) = d(200 − 30).
- Compare the two final prices.
Final answer
Apply the discount first: $130 versus $136. The 20% comes off a larger amount, so taking the percentage off first saves more.
IB-style question — composite from a table
A function is given by a table: g(1) = 4, g(4) = 2, g(2) = 5, and f(2) = 7, f(5) = 1, f(4) = 3.
Find (f∘g)(1) and (f∘g)(2).
Step by step
- (f∘g)(1): inner first, g(1) = 4, then f(4).
- (f∘g)(2): g(2) = 5, then f(5).
Final answer
(f∘g)(1) = 3 and (f∘g)(2) = 1 — read g first, then look that output up in f.