Decomposing Functions (College Board AP® Precalculus): Study Guide

Decomposing functions

What does it mean to decompose a function?

• Functions given analytically can often be decomposed (broken down) into less complicated functions

  • This means rewriting a single function

    • as the composition of two or more simpler functions

  • E.g. the function $h\left(x\right)=(2x+1{)}^3$ can be decomposed as

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

      • where $f\left(x\right)=x^3$

      • and $g\left(x\right)=2x+1$

• You can check a decomposition is correct by substituting the inner function back into the outer function

  • If every $x$ in the outer function gets replaced by the entire inner function

    • then the result should match the original

  • In the example above

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

    • so $f\left(g\right(x\left)\right)=(2x+1{)}^3$

    • this matches $h\left(x\right)$ ✓

How do you find a decomposition?

• Look for a structure where one operation is applied to the result of another

  • Start by identifying an "inner" part of the expression

    • this becomes $g\left(x\right)$

  • To find the outer function $f$

    • Take the original expression and swap out every instance of the inner part for a plain $\mathit{x}$

    • What remains is $f\left(x\right)$

• E.g. decompose $h\left(x\right)=\sqrt{x^2-5}$

  • The inner part is $x^2-5$

    • so let $g\left(x\right)=x^2-5$

  • The outer operation is taking the square root

    • so let $f\left(x\right)=\sqrt{x}$

  • Then $h\left(x\right)=f\left(g\right(x\left)\right)=\sqrt{x^2-5}$ ✓

• Decompositions are not unique

  • The same function can sometimes be decomposed in different ways

  • E.g. $h\left(x\right)=(x+3{)}^2+5$ could be decomposed as $f\left(g\right(x\left)\right)$

    • with $f\left(x\right)=x^2+5$ and $g\left(x\right)=x+3$

    • or with $f\left(x\right)=x+5$ and $g\left(x\right)=(x+3{)}^2$

How are transformations related to compositions?

• Vertical and horizontal translations of a function $f$ correspond to additive transformations

  • These can be understood as compositions of $f$ with $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{x}\mathbf{+}\mathit{k}$

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

      • That is a horizontal translation by $k$ units (to the left if $k$ is positive, to the right if $k$ is negative)

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

      • That is a vertical translation up by $k$ units (up if $k$ is positive, down if $k$ is negative)

• Vertical and horizontal dilations of a function $f$ correspond to multiplicative transformations

  • These can be understood as compositions of $f$ with $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{k}\mathit{x}$

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

      • That is a horizontal dilation by a factor of $\frac{1}{k}$

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

      • That is a vertical dilation by a factor of $k$

• This means that every transformation you have studied can be thought of as a composition of functions