Composite Functions (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Composite functions

What is a composite function?

• A composite function is where one function is applied after another function

• The ‘output’ of one function will be the ‘input’ of the next one

• Sometimes called function-of-a-function

• A composite function can be denoted

  • $\mathrm{fg}\left(x\right)$

  • $\mathrm{f}\left(\mathrm{g}\right(x\left)\right)$

  • All of these mean “$\mathrm{f}$ of $\mathrm{g}\left(x\right)$”

How do I work with composite functions?

• Recognise the notation

  • $\mathbf{fg}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ means “f of g of x”

  The order matters

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

  • Then apply $\mathrm{f}$ to $\mathrm{g}\left(x\right)$ to get $\mathrm{f}\left(\mathrm{g}\right(x\left)\right)$

  • Always start with the function closest to the variable

  • $\mathrm{fg}\left(x\right)$ is not usually equal to $\mathrm{gf}\left(x\right)$

Special cases

• $\mathrm{fg}\left(x\right)$ and $\mathrm{gf}\left(x\right)$ are generally different but can sometimes be the same

• $\mathrm{ff}\left(x\right)$ is written as ${\mathrm{f}}^2\left(x\right)$

  • Note that trig functions are exceptions to this rule

    • e.g. ${\sin }^2\left(x\right)$ means ${\left(\sin \left(x\right)\right)}^2$ not $\sin \left(\sin \left(x\right)\right)$

• For inverse functions, ${\mathrm{ff}}^{-1}\left(x\right)={\mathrm{f}}^{-1}\mathrm{f}\left(x\right)=x$

Domain & range of composite functions

How do I find the domain and range of composite functions?

• Use logic to determine the domain and range of a composite function

• For $\mathrm{fg}\left(x\right)$ the first function to be applied will be $\mathrm{g}$

  • So, at best, the domain of $\mathrm{fg}\left(x\right)$ will be the same as the domain of $\mathrm{g}\left(x\right)$

• However, for this to be the case, the range of $\mathrm{g}\left(x\right)$ must be contained within the domain of $\mathrm{f}\left(x\right)$

  • If this is not the case, then restrictions on the domain of $\mathrm{fg}\left(x\right)$ will be required

• Similarly, at best, the range of $\mathrm{fg}\left(x\right)$ will be the same as the range of $\mathrm{f}\left(x\right)$

  • But if the domain of $\mathrm{f}\left(x\right)$ has been affected, the range of $\mathrm{fg}\left(x\right)$ will also be affected