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## Subsection1.5Function Notation

There is a convenient notation for discussing functions. First, we choose a letter, such as $f\text{,}$ $g\text{,}$ or $h$ (or $F\text{,}$ $G\text{,}$ or $H$), to name a particular function. (We can use any letter, but these are the most common choices.) For instance, the height, $h\text{,}$ of a falling textbook is a function of the elapsed time, $t\text{.}$ We might call this function $f\text{.}$ In other words, $f$ is the name of the relationship between the variables $h$ and $t\text{.}$ We write

\begin{equation*} h = f (t) \end{equation*}

which means "$h$ is a function of $t\text{,}$ and $f$ is the name of the function."

###### Example1.11

In Example1.10, the height of an algebra book dropped from the top of One World Trade Center is given by the equation

\begin{equation*} h = 1776 - 16t^2 \end{equation*}

We see that

 when $t=1$ $h=1760$ when $t=2$ $h=1712$

Using function notation, these relationships can be expressed more concisely as

 $f(1)=1760$ and $f(2)=1712$

which we read as "$f$ of 1 equals 1760" and "$f$ of 2 equals 1712." The values for the input variable, $t\text{,}$ appear inside the parentheses, and the values for the output variable, $h\text{,}$ appear on the other side of the equation.

Remember that when we write $y = f(x)\text{,}$ the symbol $f(x)$ is just another name for the output variable.