Composite Functions (Edexcel GCSE Maths: Higher): Revision Note

Composite functions

What is a composite function?

• A composite function is a function applied to the output of another function

  • The input goes through the 1st function to become an output

  • This output goes through the 2nd function to become a new output

What notation is used for composite functions?

• If $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ are two functions, then

  • $\mathrm{g}\left(\mathrm{f}\left(x\right)\right)$ is a composite function

  • It means the input $x$goes through function $\mathrm{f}$ first

    • This gives the output $\mathrm{f}\left(x\right)$

    • Then this output, $\mathrm{f}\left(x\right)$, becomes the input of function $\mathrm{g}$, giving $\mathrm{g}\left(\mathrm{f}\left(x\right)\right)$

  • $\mathrm{gf}\left(x\right)$ is the shorthand notation used for $\mathrm{g}\left(\mathrm{f}\left(x\right)\right)$

    • It means do $\mathrm{f}$ first, then $\mathrm{g}$

    • The order of applying the functions goes from right to left

    • (the letter nearest the bracket goes first)

    • This is often the opposite of what people expect!

  • $\mathrm{fg}\left(x\right)$ means do $\mathrm{g}\left(x\right)$ first then $\mathrm{f}\left(x\right)$ second

  • $\mathrm{ff}\left(x\right)$ means apply $\mathrm{f}\left(x\right)$ twice!

    • This can be written ${\mathrm{f}}^2\left(x\right)$

    • This does not mean the same as ${\left[\mathrm{f}\left(x\right)\right]}^2$

How do I substitute numbers into composite functions?

• If you are putting a number into a composite function

  • put the number into the function closest to (x)

  • then make the output of the first function the input of the second function

• For example, if $\mathrm{f}\left(x\right)=2x+1$ and $\mathrm{g}\left(x\right)=\frac{1}{x}$

  • to find $\mathrm{gf}\left(2\right)$:

    • Put the 2 in as the input of $\mathrm{f}$ first

    • $\mathrm{f}\left(2\right)=2\left(2\right)+1=5$

    • Then put 5 in as the input of $\mathrm{g}$

    • So $\mathrm{gf}\left(2\right)=\mathrm{g}\left(\mathrm{f}\left(2\right)\right)=\mathrm{g}\left(5\right)=\frac{1}{5}$

  • to find $\mathrm{fg}\left(2\right)$:

    • Put the 2 in as the input of $\mathrm{g}$ first

    • $\mathrm{g}\left(2\right)=\frac{1}{2}$

    • Then put $\frac{1}{2}$ in as the input of $\mathrm{f}$

    • So $\mathrm{fg}\left(2\right)=\mathrm{f}\left(\mathrm{g}\left(2\right)\right)=\mathrm{f}\left(\frac{1}{2}\right)=2\left(\frac{1}{2}\right)+1=2$

  • to find $\mathrm{ff}\left(2\right)$:

    • $\mathrm{f}\left(2\right)=2\times 2+1=5$

    • $\mathrm{f}\left(5\right)=2\times 5+1=11$

    • so $\mathrm{ff}\left(2\right)=11$

How do I find composite functions algebraically?

• If you are using algebra, substitute the whole algebraic expression as your input

  • For example, if $\mathrm{f}\left(x\right)=2x+1$ and $\mathrm{g}\left(x\right)=\frac{1}{x}$

    • $\mathrm{fg}\left(x\right)=\mathrm{f}\left(\mathrm{g}\left(x\right)\right)=\mathrm{f}\left(\frac{1}{x}\right)=2\times \left(\frac{1}{x}\right)+1=\frac{2}{x}+1$

    • $\mathrm{gf}\left(x\right)=\mathrm{g}\left(\mathrm{f}\left(x\right)\right)=\mathrm{g}(2x+1)=\frac{1}{2x+1}$

    • $\mathrm{ff}\left(x\right)=\mathrm{f}\left(\mathrm{f}\left(x\right)\right)=\mathrm{f}\left(2x+1\right)=2\left(2x+1\right)+1$ which simplifies to $\mathrm{ff}\left(x\right)=4x+3$