5.02 Function notation
We have seen that functions are special types of relations. When dealing with functions we often use function notation to emphasise that we are dealing with a special case, to highlight the independent and dependent variable and for ease of notation with substitution and calculus.
When we are writing in function notation, instead of writing "$y=$y=", we write "$f(x)=$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 we can use variables that have meaning in context.
Exploration
Instead of $y=2x+1$y=2x+1 we could write $f(x)=2x+1$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.
$P(t)=200\times0.8^t$P(t)=200×0.8t, here population is a function of time.
$H(d)=30-2d-30d^2$H(d)=30−2d−30d2, here height is a function of distance.
These are all examples of just a different way to write '$y=$y= …' notation as a function. The letter in the bracket on the left is the input and the right hand side gives us a rule for the output.
Function notation also allows for shorthand for substitution. If $y=3x+2$y=3x+2 and we write this in function notation as $f(x)=3x+2$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 $Q(x)=x^2+9x$Q(x)=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 |
(c) $A(p)$A(p)
Think: This means we need to substitute $p$p in for $x$x in the $A(x)$A(x) equation.
Do:
| $A(p)$A(p) | $=$= | $p^2+1$p2+1 |
Practice questions
Question 1
Consider the equation $x-4y=8$x−4y=8.
Assume that $y$y is a function of $x$x.
Rewrite the equation using function notation $f\left(x\right)$f(x).
Find the value of $f\left(12\right)$f(12).
Question 2
Consider the function $f\left(x\right)=2x^3+3x^2-4$f(x)=2x3+3x2−4.
Evaluate $f\left(0\right)$f(0).
Evaluate $f\left(\frac{1}{4}\right)$f(14).
Question 3
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.
Question 4
Consider the function $g\left(x\right)=ax^3-3x+5$g(x)=ax3−3x+5.
Determine $g\left(k\right)$g(k).
Form an expression for $g\left(-k\right)$g(−k).
Is $g\left(k\right)=g\left(-k\right)$g(k)=g(−k)?
Yes
ANo
BIs $g\left(k\right)=-g\left(-k\right)$g(k)=−g(−k)?
Yes
ANo
B