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

Subsection1.2Functions Defined by Tables

When we use a table to describe a function, the first variable in the table (the left column of a vertical table or the top row of a horizontal table) is the input variable, and the second variable is the output. We say that the output variable is a function of the input.

Example1.1

  1. Table1.2 shows data on sales compiled over several years by the accounting office for Eau Claire Auto Parts, a division of Major Motors. In this example, the year is the input variable, and total sales is the output. We say that total sales, \(S\text{,}\) is a function of \(t\text{.}\)

    Year \((t)\) Total sales \((S)\)
    2000 $612,000
    2001 $663,000
    2002 $692,000
    2003 $749,000
    2004 $904,000
    Table1.2

  2. Table1.3 gives the cost of sending printed material by first-class mail in 2016.

    Weight in ounces \((w)\) Postage \((P)\)
    \(0 \lt w \le 1 \) $0.47
    \(1 \lt w \le 2 \) $0.68
    \(2 \lt w \le 3 \) $0.89
    \(3 \lt w \le 4 \) $1.10
    \(4 \lt w \le 5 \) $1.31
    \(5 \lt w \le 6 \) $1.52
    \(6 \lt w \le 7 \) $1.73
    Table1.3

    If we know the weight of the article being shipped, we can determine the required postage from Table1.3. For instance, a catalog weighing 4.5 ounces would require $1.31 in postage. In this example, \(w\) is the input variable and \(p\) is the output variable. We say that \(p\) is a function of \(w\text{.}\)

  3. Table1.4 records the age and cholesterol count for 20 patients tested in a hospital survey.

    Age Cholesterol count Age Cholesterol count
    53 217 \(51\) \(209\)
    48 232 53 241
    55 198 49 186
    56 238 \(51\) \(216\)
    \(51\) \(227\) 57 208
    52 264 52 248
    53 195 50 214
    47 203 56 271
    48 212 53 193
    50 234 48 172
    Table1.4

    According to these data, cholesterol count is not a function of age, because several patients who are the same age have different cholesterol levels. For example, three different patients are 51 years old but have cholesterol counts of 227, 209, and 216, respectively. Thus, we cannot determine a unique value of the output variable (cholesterol count) from the value of the input variable (age). Other factors besides age must influence a persons cholesterol count.