How Do You Find a Composite Function f(g(x)) in a Real-World Context?

You find a composite function by using the output of the inner function, $g$, as the input of the outer function, $f$. Therefore, (f(g(x))) means apply $g$ first, then apply $f$.

The Method

A composite function combines two processes into one. In a real-world context, identify what each function does and check that the output units of $g$ match the required input units of $f$.

Suppose a delivery company calculates the mass of a parcel from its volume using

$$g(x)=0.4x+2,$$

where $x$ is the parcel's volume in litres and $g(x)$ is its mass in kilograms. The delivery cost for a parcel of mass $m$ kilograms is

$$f(m)=3m+5.$$

To find the cost directly from the volume, substitute the complete expression for $g(x)$ into $f$:

$$\begin{aligned} f(g(x))&=3g(x)+5\\ &=3(0.4x+2)+5\\ &=1.2x+11. \end{aligned}$$

Thus, $f(g(x))=1.2x+11$ gives the delivery cost from a parcel volume of $x$ litres. For example, if $x=10$, then

$$f(g(10))=1.2(10)+11=23.$$

The delivery cost is therefore 23 currency units.

The order is determined by the flow of quantities through the model. In this example, volume is converted to mass, then mass is converted to cost. The reverse composition would require a cost to be entered into a function designed for volume, so it has no meaningful interpretation here.

A useful check is to evaluate the inner function first: $g(10)=6$, then $f(6)=23$. This must match the simplified rule.

The domain must also make contextual sense. Here, volume cannot be negative, so a suitable domain is $x\geq 0$. More formally, $x$ must be in the domain of $g$, and $g(x)$ must lie in the domain of $f$.

A common misconception is that $f(g(x))=g(f(x))$. This is generally false because changing the order changes which process happens first.

Exam Technique

This is AHL 2.7: Composite functions, so it is HL-only content and may appear in any GDC-active Math AI HL paper. Show the substitution explicitly, simplify accurately, state the contextual meaning and units, and check whether the composite function requires a restricted domain.