Constructing Composite Functions (College Board AP® Precalculus): Study Guide

Constructing composite functions

Why is function composition useful?

• A composition of functions can be used to relate two quantities that are not directly related by an existing formula

• E.g. if one function relates time to temperature

  • and another relates temperature to pressure

  • then composing them gives a function that relates time to pressure directly

How do I construct a composite function analytically?

• When analytical (formula) representations of the functions $f$ and $g$ are available

  • You can construct an analytical representation of $f\left(g\right(x\left)\right)$

  • by substituting $g\left(x\right)$ for every instance of $x$ in the formula for $f$

• To do this

  • Start by writing out the expression for $f\left(x\right)$

  • Then replace every $x$ in that expression with the entire expression for $g\left(x\right)$

  • Finally, simplify the resulting expression

• E.g. if $f\left(x\right)=3x+1$ and $g\left(x\right)=x^2-4$

  • Write out the expression for $f\left(x\right)$

    • $f\left(x\right)=3x+1$

  • Substitute the expression for $g\left(x\right)$ everywhere that $x$ appears

    • $f\left(g\right(x\left)\right)=3(x^2-4)+1$

  • Expand the brackets and simplify

    • $f\left(g\right(x\left)\right)=3x^2-12+1=3x^2-11$

• Be careful to substitute $g\left(x\right)$ for every occurrence of $x$ in $f$

  • E.g. if $f\left(x\right)=x^2+2x$ and $g\left(x\right)=x-3$

    • $f\left(g\right(x\left)\right)=(x-3{)}^2+2(x-3)=x^2-6x+9+2x-6=x^2-4x+3$

  • Both the $x^2$ and $2x$ terms required substitution

How do you construct a composite function numerically or graphically?

• A numerical representation (table) of $f\circ g$ can often be constructed by calculating pairs of values $(x,f(g\left(x\right)\left)\right)$

  • I.e. for each input $x$, find $g\left(x\right)$, then find $f$ at that value

  • Record the pairs $(x,f(g\left(x\right)\left)\right)$ to build a table for the composite function

• A graphical representation of $f\circ g$ can be constructed similarly

  • For selected input values $x$, calculate or estimate $f\left(g\right(x\left)\right)$

  • Plot the resulting points $(x,f(g\left(x\right)\left)\right)$

  • Connect or sketch the graph based on the plotted points

• These approaches are especially useful when one or both functions are given as tables or graphs rather than formulas