Function Definitions

For example, the rule ‘double the number and add 1’ takes an input of 3 and produces $2\times3+1=7$. An input of $-4$ produces $2\times(-4)+1=-7$. The rule stays the same even though the input changes.

The important condition is that a particular input must give just one output. Different inputs can give the same output, as we will see with a squaring rule.

Reading function notation

We can name a function using a letter, usually $f$, $g$ or $h$. The rule ‘double the number and add 1’ can be written as

Read $f(x)$ as ‘f of x’. Here, $f$ names the function, $x$ represents the input, and $f(x)$ represents the output. The brackets show what goes into the function; they do not mean that $f$ is multiplied by $x$.

So $f(3)$ means ‘the output of function $f$ when the input is 3’. For this rule, $f(3)=7$.

Writing $y=f(x)$ simply uses $y$ as another name for the output. A definition such as $g(x)=3x-4$ names a different function, with its own rule: triple the input and subtract 4.

Evaluating a function

To evaluate a function, substitute the given input everywhere the input variable appears, then calculate using the usual order of operations.

Consider the function

Its rule is ‘square the input, then subtract 9’. For an input of 5,

The statement $f(5)=16$ tells us that 5 is the input and 16 is the output.

For a negative input, brackets show that the whole negative number is being squared:

Also, $f(3)=3^2-9=0$. Both inputs produce 0, but this is still a function: each input produces exactly one output.

Some rules contain the input more than once. For example, if $h(x)=3x^2-2x+1$, then

Both occurrences of $x$ have been replaced by 7. The square is calculated before the multiplication and subtraction.

An expression can be the input

The input need not be a single number. Whatever is inside the function brackets replaces the input variable in the rule.

For $f(x)=x^2-9$, an input of $a$ gives $f(a)=a^2-9$. An input of $a+1$ gives

The whole expression $a+1$ is squared because it is the whole input. By contrast, $f(a)+1$ means apply the function to $a$ first, then add 1 to its output:

The position of the brackets therefore changes which calculation is being requested.