Skip to main content
\(\newcommand{\dollar}{\$} \DeclareMathOperator{\erf}{erf} \DeclareMathOperator{\arctanh}{arctanh} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

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."


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.