Function Notation f(x) (SQA National 5 Maths): Revision Note

Using functional 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 -5 in would give 2 × (-5) + 1 = -9 

    • Putting $x$ in would give $2x+1$

What is function notation?

• A function, $f$, with input $x$ can be written as $f\left(x\right)=...$

  • 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

  • $f\left(x\right)=3x–4$ 

• In such cases, "$x$" is the input and "$f\left(x\right)$" is the output

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

  • E.g. $y=3x–4$

How does a function work?

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

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

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

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

    • You cannot simplify this output any further

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

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

    • $f\left(3\right)=2\times 3+1=7$

    • $f(-4)=2\times (-4)+1=-7$

    • $f\left(a\right)=2a+1$

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

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

    • If $f\left(x\right)=15$, then form an equation by replacing $\mathit{f}\left(x\right)$ with $2x+1$

    • $2x+1=15$

    • Solving this equation gives an input of $x=7$

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

  • $f\left(x\right)=15$ means an input of $x$ gives an output of 15

  • $f\mathit{\left(}15\mathit{\right)}$ means substitute the input 15 into the function