Function Notation

Once the machines in The Function Machine earned names, we needed a way to talk about them without paragraphs. That is all $f(x)$ is: the machine’s name, with its input shown in brackets. Read it aloud as ”$f$ of $x$” — it is a name and an address, never multiplication.

Reading the notation

Written	Say it	It means
$f(x)=3x-5$	”$f$ of $x$ is $3x-5$“	the rule itself
$f(4)$	”$f$ of 4”	the output when 4 goes in — here, $7$
$f(4)=7$	—	the point $(4,7)$ sits on the graph of $f$
$f(x)=0$	—	a question: which inputs give output 0?

The last two rows are where the notation earns its keep. A statement like $h(3)=25$ compresses a whole sentence — “three seconds after launch, the height was 25 metres” — into six characters, and it tells you a point on the graph at the same time.

Evaluating without fear

To evaluate, wrap the input in brackets before touching anything else. For $f(x)=2x^2+3x-1$:

$$f\left(\frac{1}{2}\right)=2{\left(\frac{1}{2}\right)}^2+3\left(\frac{1}{2}\right)-1=\frac{1}{2}+\frac{3}{2}-1=1$$

Now try $f(-2)$: $2(4)-6-1=1$. Two different inputs, the same output — and no rule broken, as What Is a Function explains. If your groupmate gets your answer from a different input, nobody has necessarily erred; compare the substitution lines, not the final numbers.

When a quadratic wears function notation in an application — a profit function $P(x)$, a height function $h(t)$ — every question in the problem becomes a notation statement: “the maximum profit” asks for the largest value $P(x)$ reaches, and “when does it hit the ground” asks you to solve $h(t)=0$. Translating the sentence into notation is half the solution.

A statement worth arguing about in True or False: is $f(2+1)=f(2)+f(1)$? Test it on the $f$ above before you vote. Then head to Function Notation Practice to make the moves automatic.

Curriculum connection

A1.2

represent linear and quadratic functions using function notation, given their equations, tables of values, or graphs, and substitute into and evaluate functions [e.g., evaluate $f\left(\frac{1}{2}\right)$, given $f(x)=2x^2+3x-1$]

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A2.3

solve problems involving quadratic functions arising from real-world applications and represented using function notation Sample problem: The profit, $P(x)$, of a video company, in thousands of dollars, is given by $P(x)=-5x^2+550x-5000$, where $x$ is the amount spent on advertising, in thousands of dollars. Determine the maximum profit that the company can make, and the amounts spent on advertising that will result in a profit and that will result in a profit of at least $4 000 000.

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