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.
The table below lists the input number of each month (January = [latex]1[/latex], February = [latex]2[/latex], 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 [latex]f[/latex] where [latex]D=f(m)[/latex] identifies months by an integer rather than by name.
| Month number, [latex]m[/latex] (input) | [latex]1[/latex] | [latex]2[/latex] | [latex]3[/latex] | [latex]4[/latex] | [latex]5[/latex] | [latex]6[/latex] | [latex]7[/latex] | [latex]8[/latex] | [latex]9[/latex] | [latex]10[/latex] | [latex]11[/latex] | [latex]12[/latex] |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Days in month, [latex]D[/latex] (output) | [latex]31[/latex] | [latex]28[/latex] | [latex]31[/latex] | [latex]30[/latex] | [latex]31[/latex] | [latex]30[/latex] | [latex]31[/latex] | [latex]31[/latex] | [latex]30[/latex] | [latex]31[/latex] | [latex]30[/latex] | [latex]31[/latex] |
The table below displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, [latex]5[/latex] years, has two different output values, [latex]40[/latex] in. and [latex]42[/latex] in.
| Age in years, [latex]a[/latex] (input) | [latex]5[/latex] | [latex]5[/latex] | [latex]6[/latex] | [latex]7[/latex] | [latex]8[/latex] | [latex]9[/latex] | [latex]10[/latex] |
|---|---|---|---|---|---|---|---|
| Height in inches, [latex]h[/latex] (output) | [latex]40[/latex] | [latex]42[/latex] | [latex]44[/latex] | [latex]47[/latex] | [latex]50[/latex] | [latex]52[/latex] | [latex]54[/latex] |
- Identify the input and output values.
- Check to see if each input value is paired with only one output value. If so, the table represents a function.
Function Notation
Some people think of functions as “mathematical machines.” Imagine you have a machine that changes a number according to a specific rule such as “multiply by [latex]3[/latex] and add [latex]2[/latex]” or “divide by [latex]5[/latex], add [latex]25[/latex], and multiply by [latex]−1[/latex].” If you put a number into the machine, a new number will pop out the other end having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.
You can also call the machine “[latex]f[/latex]” for function. If you put [latex]x[/latex] into the box, [latex]f(x)[/latex], comes out. Mathematically speaking, [latex]x[/latex] is the input, or the “independent variable,” and [latex]f(x)[/latex] is the output, or the “dependent variable,” since it depends on the value of [latex]x[/latex].
function notation
The notation [latex]y=f(x)[/latex] defines a function named [latex]f[/latex]. This is read as “[latex]y[/latex] is a function of [latex]x[/latex].” The letter [latex]x[/latex] represents the input value, or independent variable. The letter [latex]y[/latex], or [latex]f(x)[/latex], represents the output value, or dependent variable.
[latex]f(x)=4x+1[/latex] is written in function notation and is read “[latex]f[/latex] of [latex]x[/latex] equals [latex]4x[/latex] plus [latex]1[/latex].” It represents the following situation: A function named [latex]f[/latex] acts upon an input, [latex]i[/latex] and produces [latex]f(x)[/latex] which is equal to [latex]4x+1[/latex]. This is the same as the equation [latex]y=4x+1[/latex].
Function notation gives you more flexibility because you do not have to use y for every equation. Instead, you could use [latex]f(x)[/latex] or [latex]g(x)[/latex] or [latex]c(x)[/latex]. This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.
To represent “height is a function of age,” we start by identifying the descriptive variables [latex]h[/latex] for height and [latex]a[/latex] for age. The letters [latex]f,g,[/latex] and [latex]h[/latex] are often used to represent functions just as we use [latex]x,y,[/latex] and [latex]z[/latex] to represent numbers and [latex]A,B,[/latex] and [latex]C[/latex] to represent sets.
[latex]\begin{array}{ll} h \text{ is } f \text{ of } a & \text{We name the function } f; \text{ height is a function of age.} \\ h = f(a) & \text{We use parentheses to indicate the function input.} \\ f(a) & \text{We name the function } f; \text{ the expression is read as "} f \text{ of } a." \end{array}[/latex]
Remember, we can use any letter to name the function; the notation [latex]h(a)[/latex] shows us that [latex]h[/latex] depends on [latex]a[/latex]. The value [latex]a[/latex] must be put into the function [latex]h[/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.
Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.