Remember that a function is a special type of relation, where each input only has one output. When dealing with functions, function notation is used to highlight the independent and dependent variables, for ease of notation with substitution, and to express functions in a standardised way that is easier to read and understand.

When writing in function notation, instead of writing "$y=$y=", write "$f(x)=$f(x)=" or $f:x\longrightarrow$f:x→. This can be interpreted as "$f$f is a function of the variable $x$x" and read as "$f$f of $x$x". Common letters to use for general functions are lower case $f$f, $g$g and $h$h. However, any letter can be used and often, variables are chosen that have meaning in context such as $t$t for time or $d$d for distance.

Exploration

Instead of $y=2x+1$y=2x+1 we could write $f(x)=2x+1$f(x)=2x+1 or $f:x\longrightarrow2x+1.$f:x→2x+1.Here $f$f is a function of the variable $x$x which follows the rule double $x$x and add $1$1.

Some examples of other functions written in function notation are:

$P(t)=200\times0.8^t$P(t)=200×0.8t, where population is a function of time.

$H(d)=30-2d-30d^2$H(d)=30−2d−30d2, where height is a function of distance.

In these examples, the letter in the bracket on the left is the input, and the right-hand side gives the rule for the output.

Function notation also allows for shorthand for substitution. If $y=3x+2$y=3x+2 and this is written in function notation as $f(x)=3x+2$f(x)=3x+2 or $f:x\longrightarrow3x+2$f:x→3x+2 then the question 'what is the value of $y$y when $x$x is $5$5?' can be asked simply as 'what is $f(5)$f(5)?' or 'Evaluate $f(5)$f(5).'

Worked example

If $A(x)=x^2+1$A(x)=x2+1 and$Q:x\longrightarrow$Q:x→ $x^2+9x$x2+9x, find:

A) $A(5)$A(5)

Think: This means we need to substitute $5$5 in for $x$x in the $A(x)$A(x) equation.

Do:

$A(5)$`A`(5)	$=$=	$5^2+1$52+1
	$=$=	$26$26

B) $Q:6$Q:6

Think: This means we need to substitute $6$6 in for $x$x in the $Q(x)$Q(x) equation.

Do:

$Q:6$`Q`:6	$=$=	$6^2+9\times6$62+9×6
	$=$=	$36+54$36+54
	$=$=	$90$90

Note: The interactive questions focus on the $f(x)$f(x) notation. The worksheet will have a combination of both function notations.

Practice questions

Question 1

Consider the function $f\left(x\right)=8x+6$f(x)=8x+6.

Determine the output produced by the input value $x=5$x=5.

Question 2

If $Z(y)=y^2+12y+32$Z(y)=y2+12y+32, find $y$y when $Z(y)=-3$Z(y)=−3.

Write both solutions on the same line separated by a comma.

Outcomes

0580E2.9

f: x ⟼ 3x – 5, to describe simple functions. Find inverse functions f ^(–1)(x). Form composite functions as defined by gf(x) = g(f(x)).

21.02 Function notation (Extended)