Composite Functions (Edexcel IGCSE Maths)

Spec point — 3.3I

Composite functions

What is a composite function?

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

What notation is used for composite functions?

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

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

    • Also written $\mathrm{gf}:x$

  • It means the input $x$goes through function$f$first

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

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

    • It means do $f$ first, then $g$
    • 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!

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

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

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

Exam Hint A good trick in the exam is to write brackets around $gf ( x )$ to make it $\mathrm{g} ( \mathrm{f} ( x ) )$, to see that it is "g" of "f(x)".

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 $\mathrm{f} ( x ) = 2 x + 1$ and $\mathrm{g} ( x ) = \frac{1}{x}$

  • to find $gf ( 2 )$:

    • Put the 2 in as the input of $f$ first
    • $\mathrm{f} ( 2 ) = 2 ( 2 ) + 1 = 5$
    • Then put 5 in as the input of $g$
    • So $\text{gf} ( 2 ) = \mathrm{g} ( \mathrm{f} ( 2 ) ) = \mathrm{g} ( 5 ) = 1 \frac{1}{5}$

  • to find $fg ( 2 )$:

    • Put the 2 in as the input of $g$ first
    • $\mathrm{g} ( 2 ) = 1 \frac{1}{2}$
    • Then put $\frac{1}{2}$ in as the input of $f$
    • So $\text{fg} ( 2 ) = \mathrm{f} ( \mathrm{g} ( 2 ) ) = \mathrm{f} ( 1 \frac{1}{2} ) = 2 ( 1 \frac{1}{2} ) + 1 = 2$

  • to find $ff ( 2 )$:

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

How do I find composite functions algebraically?

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

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

    • $\text{fg} ( x ) = \mathrm{f} ( \mathrm{g} ( x ) ) = \mathrm{f} ( \frac{1}{x} ) = 2 \times ( \frac{1}{x} ) + 1 = \frac{2}{x} + 1$
    • $\text{gf} ( x ) = \mathrm{g} ( \mathrm{f} ( x ) ) = \mathrm{g} ( 2 x + 1 ) = \frac{1}{2 x + 1}$
    • $\text{ff} ( x ) = \mathrm{f} ( \mathrm{f} ( x ) ) = \mathrm{f} ( 2 x + 1 ) = 2 ( 2 x + 1 ) + 1$ which simplifies to $\text{ff} ( x ) = 4 x + 3$

Worked Example In this question, $\mathrm{f} ( x ) = 2 x - 1$ and $\mathrm{g} : x \rightarrow ( x + 2 )^{2}$.

(a) Find $fg ( 4 )$.

Answer:

"g" is on the inside of the composite function, so apply g first

$\mathrm{g} ( 4 ) = ( 4 + 2 )^{2} = 6^{2} = 36$

Now apply the function "f" to 36

$\begin{gathered} \mathrm{f} ( 36 ) & = 2 ( 36 ) - 1 \\ = 72 - 1 \end{gathered}$

$\mathrm{fg}(4)=71$

(b) Find $gf ( x )$.

Answer:

"f" is on the inside of the composite function so substitute the function f(x) into g(x) It can help to write $\text{gf} ( x ) = \mathrm{g} ( \mathrm{f} ( x ) )$

• *$\text{gf} ( x ) = \mathrm{g} ( \mathrm{f} ( x ) ) = \mathrm{g} ( 2 x - 1 ) = ( ( 2 x - 1 ) + 2 )^{2}$

Simplify inside the bracket

$\begin{gathered} \text{gf} ( x ) & = ( 2 x - 1 + 2 )^{2} \end{gathered}$

$\mathrm{gf}(x)=(2x+1)^{2}$