Function Notation

Sommaire

In short: Function notation writes a function as f(x), where f is the name of the rule and x is the input, so f(5) means the output when the input is 5.

Algebra 1Functions and GraphsF-IF.A.2

What is Function Notation?

Picture a vending machine, one button and one snack: function notation is the label on that machine. In $f(x) = 2x + 3$, the letter f names the rule, the x in parentheses is the input, and the right side tells what to do with it.

The symbol $f(4)$ is read “f of 4” and means replace every x with 4: $f(4) = 2(4) + 3 = 11$. It is not a multiplication of f and 4. Different letters, such as g or C, name different functions.

How to spot it

You see f(x), g(t) or C(n) in a problem instead of y.
A question asks you to find f(5), or to find x when f(x) equals a given number.
A real situation uses function notation for a named rule, like cost as a function of the number of items.
An expression such as f(a + 2) asks for a substitution of something other than a number.

The method

Read the rule and locate the input variable inside the parentheses.
Replace every copy of that variable with the value or expression asked for, using parentheses.
Simplify with the order of operations to get the output.
To go backward, set the rule equal to the given output and solve for the input.

f(4) = 2(4) + 3 = 11: the input 4 becomes the output 11.

Function Notation: solved example

Problem

A print shop charges $C(n) = 12.5n + 40$ dollars for n custom T-shirts. Find C(18), and find how many shirts cost $177.50.

Solution

$C(18) = 12.5(18) + 40 = 225 + 40 = 265$. Eighteen shirts cost $265.

For the reverse, solve $12.5n + 40 = 177.5$: $12.5n = 137.5$, so $n = 11$. An order of 11 shirts costs $177.50.

Try it

Question 1

If $g(x) = x^2 – 3x$, find $g(-2)$.

Show the answer

$(-2)^2 – 3(-2) = 4 + 6 = 10$.

Question 2

If $f(x) = 5x + 1$, what is $f(a + 2)$?

Show the answer

$5(a + 2) + 1 = 5a + 10 + 1 = 5a + 11$.

Function Notation vs. Function

A function is the idea: a rule giving exactly one output for each input. Function notation, the f(x) writing, is the language used to name that rule and to ask about specific inputs. Writing f(3) does not multiply f by 3; it asks for the output of the rule at 3.

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