Composition of Functions (College Board AP® Precalculus): Study Guide

Composite functions

What is a composite function?

• If $f$ and $g$ are functions, the composite function $f\circ g$ is a function that

  • maps input values to output values

    • by using the output values of $g$ as input values for $f$

  • I.e., the input value is first 'fed into' $g$

    • then the output from $g$ is 'fed into' $f$ to find the final output of the composite function

  • $(f\circ g)\left(x\right)$ can also be written as $f\left(g\right(x\left)\right)$

    • Read this as "$f$ of $g$ of $x$"

• To find the output value of a composite function $f\left(g\right(x\left)\right)$ for a given input value

  • First find the output of the inner function $g$ at the given input $x$

    • this gives $g\left(x\right)$

  • Then use that output as the input for the outer function $f$

    • this gives $f\left(g\right(x\left)\right)$

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

    • First find $g\left(2\right)$

      • $g\left(2\right)=2^2-1=3$

    • then input that value into $f$

      • $f\left(g\left(2\right)\right)=f\left(3\right)=2\left(3\right)+3=9$

What about the domain of a composite function?

• The domain of the composite function $f\circ g$ is restricted to

  • those input values of $g$

  • for which the corresponding output value is in the domain of $f$

• If $g\left(x\right)$ produces an output that is not in the domain of $f$

  • then $f\left(g\right(x\left)\right)$ is not defined at that input

How do you evaluate a composite function from tables, graphs, or formulas?

• Values for the composite function $f\circ g$ can be calculated or estimated from graphical, numerical, analytical, or verbal representations of $f$ and $g$

• The process is always the same

  • use output values from $g$ as input values for $f$

• From a table

  • Look up $g\left(x\right)$ in the table for $g$,

    • then look up $f$ at that value in the table for $f$

  • E.g. if a table shows $g\left(2\right)=5$ and $f\left(5\right)=-3$

    • then $f\left(g\right(2\left)\right)=f\left(5\right)=-3$

• From a graph

  • Read $g\left(x\right)$ from the graph of $g$

    • then use that value as the input on the graph of $f$

• From a formula

  • Substitute the input into $g$ to get a numerical value

    • then substitute that value into $f$

• Mixed representations

  • You may need to use different representations for $f$ and $g$

  • E.g. read $g\left(x\right)$ from a table, then substitute into an analytical formula for $f$

Properties of composite functions

Does the order of composition matter?

• When the order doesn't matter for a mathematical operation, that operation is said to be commutative

  • For example, multiplication is commutative

    • $4\times 5=5\times 4=20$

    • The order in which you multiply two numbers doesn't change the answer

• The composition of functions is not commutative

  • $f\circ g$ and $g\circ f$ are generally different functions

    • So $f\left(g\right(x\left)\right)$ and $g\left(f\right(x\left)\right)$ generally have different values

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

    • $f\left(g\right(1\left)\right)=f\left(4\right)=16$

    • $g\left(f\right(1\left)\right)=g\left(1\right)=4$

    • These are not equal

      • i.e. the order matters

• Always pay close attention to which function is the inner function and which is the outer function

What is the identity function?

• The identity function is the function defined by $f\left(x\right)=x$

  • It maps every input to itself

    • the output is always equal to the input

• When the identity function is composed with any function $g$

  • $g\left(f\right(x\left)\right)=g\left(x\right)$

    • and $f\left(g\right(x\left)\right)=g\left(x\right)$

  • In both cases the result is just $g$

• The identity function acts in composition the same way that 0 acts in addition and 1 acts in multiplication

  • It leaves the other function unchanged