Functions and function notation  (Page 3/21)

We can also give an algebraic expression as the input to a function. For example $f\left(a+b\right)$ means “first add a and b , and the result is the input for the function f .” The operations must be performed in this order to obtain the correct result.

Using function notation for days in a month

Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.

The number of days in a month is a function of the name of the month, so if we name the function $f,$ we write $\text{days}=f(\text{month})$ or $d=f(m).$ The name of the month is the input to a “rule” that associates a specific number (the output) with each input.

For example, $f\left(\text{March}\right)=31,$ because March has 31 days. The notation $d=f\left(m\right)$ reminds us that the number of days, $d$ (the output), is dependent on the name of the month, $m$ (the input).

Representing functions using tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.

[link] lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function $f$ where $D=f\left(m\right)$ identifies months by an integer rather than by name.

[link] defines a function $Q=g\left(n\right).$ Remember, this notation tells us that $g$ is the name of the function that takes the input $n$ and gives the output $Q\text{\hspace{0.17em}.}$