Composite Functions (DP IB Analysis & Approaches (AA): SL): Revision Note

Composite functions

What is a composite function?

• A composite function is where a function is applied to another function

  • written as any out of

    • $(f\circ g)\left(x\right)$

    • $fg\left(x\right)$

    • $f\left(g\right(x\left)\right)$

• The order matters

  • $(f\circ g)\left(x\right)$ means:

    • First apply $g$ to $x$ to get $g\left(x\right)$

    • Then apply $f$ to the previous output to get $f\left(g\right(x\left)\right)$

  • i.e. start with the function closest to $x$

• $(f\circ g)\left(x\right)$ is not usually equal to $(g\circ f)\left(x\right)$

How do I find the domain of a composite function?

• This is best shown through an example

  • e.g. let $f\left(x\right)=2x+1,-5\le x\le 5$ and let $g\left(x\right)=\sqrt{x},1\le x\le 49$

• What is the domain of $fg\left(x\right)$?

  • In $fg\left(x\right)$, an input $x$ goes into $g\left(x\right)$ first, which accepts the inputs $1\le x\le 49$

    • i.e. the domain so far is $1\le x\le 49$

  • Substitute the inputs $1\le x\le 49$ into $g\left(x\right)$ to get the outputs $1\le g\left(x\right)\le 7$

  • The outputs $1\le g\left(x\right)\le 7$ become the new inputs for $f$

  • Substitute $1\le g\left(x\right)\le 7$ into $f$

    • Not possible: $f$ only accepts inputs between $-5$ and $5$

  • It would only work if $1\le g\left(x\right)\le 5$

    • to make it fit between $-5$ and $5$

  • but $1\le g\left(x\right)\le 5$ means the initial domain should have been $1\le x\le 25$

  • so $1\le x\le 25$ is the domain of $fg\left(x\right)$

How do I find the range of a composite function?

• Using the same example from above, to find the range of $fg\left(x\right)$

  • substitute the domain (found above) into the composite function (in order)

  • The domain is $1\le x\le 25$

    • First substitute $1\le x\le 25$ into $g\left(x\right)$ to get $1\le g\left(x\right)\le 5$

    • Then substitute $1\le g\left(x\right)\le 5$ into $f$ to get $2\times 1+1\le f\le 2\times 5+1$

    • This means the range of $fg\left(x\right)$ is $3\le fg\left(x\right)\le 11$