Advanced function modelling

Objective. Choose domains and evaluate linked function models.

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2. Worked example
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Learn: the key idea

Advanced function modelling uses composite or piecewise functions to represent real situations, requiring careful attention to each function's domain before evaluating or combining them. Example: if a delivery cost model is C(d)=5+0.8d for 0≤d≤20 and C(d)=5+0.6d for d>20, evaluating C(25) means using the second branch, giving 5+15=20. When composing functions fg(x), always check that g(x)'s output lies within f's domain before substituting further.

Worked example

A function model gives h(t)=20+3t for 0≤t≤4 and h(t)=32−2(t−4) for t>4. Find h(6) using this modelling.

1 $Check which branch applies: since t=6>4, use the second rule$
2 $Substitute t=6 into h(t)=32-2(t-4)$
3 $Compute t-4=2, so h(6)=32-2(2)$
4 $Simplify: 32-4=28$

Answer: h(6)=28

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Common mistake. Students apply the wrong branch of a piecewise model, e.g. using the first rule for d=25 above the boundary; always compare the input to the stated domain limits before selecting a formula.

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