Function Composition

Function composition sends an input through one function and then uses that output as the input of another.

A function connects an allowed input set to a resulting output set. A curve appears with a highlighted horizontal domain span and vertical range span. The two labelled spans distinguish allowed inputs from resulting outputs.

Composition builds a multi-step process by placing one function inside another.

Reading the notation

The composition of $f$ with $g$ is written

(f\circ g)(x)=f(g(x)).

Read it from right to left: apply $g$ first, then apply $f$ to the result. The circle is not multiplication. It signals that one output feeds another input.

Evaluating from values or a table

To find $(f\circ g)(2)$, first find $g(2)$. Suppose $g(2)=5$. Then look for $f(5)$. If $f(5)=-1$, the composition equals $-1$.

The intermediate value must be a valid input for $f$. A table may contain $g(2)$ but not contain the necessary row for $f(g(2))$; in that case the composition cannot be evaluated from the supplied data.

Finding a composition formula

Replace every $x$ in the outside function with the complete inside function. Parentheses are essential.

f(x)=2x+3,\qquad g(x)=x^2-1,

then

(f\circ g)(x)=f(x^2-1)=2(x^2-1)+3=2x^2+1.

Changing the order gives a different result:

(g\circ f)(x)=g(2x+3)=(2x+3)^2-1.

Composition is generally not commutative.

Worked example with careful substitution

Modelling chained relationships

Suppose a taxi distance is $d(t)=40t$ kilometres after $t$ hours, and fuel cost is $C(d)=0.18d$ dollars for distance $d$. Then

(C\circ d)(t)=0.18(40t)=7.2t.

The composite model converts hours directly to fuel cost. Tracking units through the chain helps confirm the order.

Common mistakes

Applying the left function first. In $f\circ g$, $g$ acts first.

Multiplying $f(x)g(x)$. Composition is substitution, not multiplication.

Dropping parentheses. $f(3x-2)$ requires replacing every input with the entire expression.

Assuming $f\circ g=g\circ f$. Test the two orders separately.

Ignoring the inside output restriction. A composite can have a narrower domain.