Function Notation (Cambridge International IGCSE Maths)

Function Notation

What is a function?

• A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers

• The numbers being put into the function are often called the inputs

• The numbers coming out of the function are often called the outputs

• A function may be thought of as a mathematical “machine”

  • For example, for the function “double the number and add 1”, the two mathematical operations are "multiply by 2 (×2)" and "add 1 (+1)"

    • Putting 3 in to the function would give 2 × 3 + 1 = 7
    • Putting -4 in would give 2 × (-4) + 1 = -7
    • Putting $x$ in would give $2x+1$

What is function notation?

• A function, f, with input $x$ can be written as $\mathrm{f} ( x ) =...$

  • Letters other than f can be used

    • The letters g, h and j are common but any letter can be used
    • Typically, a new letter will be used to define a new function in a question

• For example, the function with the rule “triple the number and subtract 4” would be written

  • $f(x)=3x--4$

• In such cases, "$x$" is the input and "$\mathrm{f} ( x )$" is the output

• Sometimes functions don’t have names like f and are just written as y = …

  • E.g. $y=3x--4$

How does a function work?

• A function has an input$x$ and an output$\mathrm{f} ( x )$

  • If the input is 2, then the output is $\mathrm{f} ( 2 )$

  • If the input is $m$, then the output is $\mathrm{f} ( m )$

  • If the input is $t+5$, then the output is $\mathrm{f} ( t + 5 )$

    • You cannot simplify this output any further

• If the function is known, the output can be calculated

  • Do this by substituting the value inside the brackets into the function

  • For example, given the function $f(x)=2x+1$

    • $f(3)=2\times3+1=7$
    • $f(-4)=2\times(-4)+1=-7$
    • $f(a)=2a+1$

• If the output is known, an equation can be formed and solved to find the input

  • For example, given the function $f(x)=2x+1$

    • If $f(x)=15$, then form an equation by replacing $\mathrm{f} ( x )$ with $2x+1$
    • $2x+1=15$
    • Solving this equation gives an input of 7

• Note that $\mathrm{f} ( x ) = 15$ and $\mathrm{f} ( 15 )$ are very different things:

  • $\mathrm{f} ( x ) = 15$ means an input of $x$ gives an output of 15
  • $\mathrm{f} ( 15 )$ means substitute the input 15 into the function

What is a mapping diagram?

• A mapping diagram shows a set of different inputs going into the function to become a set of different outputs

  • Transforming inputs into outputs is called mapping

• For example, a mapping diagram for the function $\mathrm{f} ( x ) = x + 3$ where $x\geq3$ could be shown as:

Worked Example A function is defined as $\mathrm{f} ( x ) = 3 x^{2} - 2 x + 1$.

Find $\mathrm{f} ( 7 )$.

The input is $x=7$, so substitute 7 into the expression everywhere you see an $x$

$\mathrm{f} ( 7 ) = 3 ( 7 )^{2} - 2 ( 7 ) + 1$* *

Calculate

$\begin{gathered} \mathrm{f} ( 7 ) & = 3 ( 49 ) - 14 + 1 \\ = 147 - 14 + 1 \end{gathered}$

$f(7)=134$

Worked Example A function is defined by $\mathrm{g} ( x ) = 3 x - 4$.

Find the value of $x$ for which $\mathrm{g} ( x ) = - 16$.

This is not saying substitute 16 into the function It says that an input $x$ is substituted into $g$ giving the output -16

To find the input, form an equation by replacing $\mathrm{g} ( x )$ with $3x-4$

$\begin{gathered} 3 x - 4 & = - 16 \end{gathered}$

Solve the equation (for example, by adding 4 to both sides, then dividing by 3)

$\begin{gathered} 3 x - 4 & = - 16 \\ 3 x & = - 12 \\ x = - \frac{12}{3} \end{gathered}$

$x=-4$