A function is a rule which maps each element of the domain to exactly one element in the range. A function can be described by many ways like with a mapping diagram, by a set of ordered pairs, by an equation, with function notation or with a function table. Each of this method is used depending upon the need.
A function table is a numerical representation of the function and is especially useful in describing the relationship in experimental data. Let us learn how to identify the functions represented by the function table by observing the pattern in the values, or how to write the function rule which the table represents.

When the tabulated values of two variables follow a rule, the table displaying these values is called a function table.

A function table consists of two columns one for input and another for output. The input column contains of values of the independent variable and the output column displays the corresponding values of the dependent variable. Each input value is paired exactly with one output value. This means, repetition of input values will not be seen.

Example:
Hours worked
x
(input)

Amount earned
in dollars y
(output)

20 250
25
300
30
350
35
400
40
450

In the above table, the input column consists of values of the independent variable x and the corresponding values of the dependent variable y are shown in the output column. This table describes y as a linear function of x.
If each input value is paired exactly with one output value, then the table describes a functional relationship between the two variables. It is possible to identify the relationship between the input and output values given in a tabular form as a commonly known function by observing the patterns in these values.

The patterns can be easily checked for Linear, quadratic and exponential functions.
Let us learn the method applied to identify a linear function table and write a function rule for the table:
  1. Find the differences between the successive input values.
  2. Find the differences between the successive output values.
  3. The table represents a linear function if the difference in output values is a constant for constant changes in the input values.
  4. This constant change in the output value for a unit difference in the input value gives the slope m of the linear function.
  5. Write the function rule as y = mx + b. The value b can be found using any pair of input - output values in the table.

Example:
Verify whether the numerical relationship shown in the table represents a function. If yes, find the function rule for the table.

Linear Table

In the above table each value of x is paired with exactly one value of y. Hence the table represents a function.

It can be seen that for every increase of +1 in the value of x, y value increases constantly by 2. Hence a linear relation is represented in the table. The slope of the linear relation is m = $\frac{2}{1}$ = 2.

Let y = 2x + b represent the function for the numerical values given in the table.
When x = 0, y = 3. Thus 2(0) + b = 3 ⇒ b = 3.
Hence the function rule for the table is y = 2x + 3.

A function table represents a Quadratic function if the second difference found for the output variables is a constant for a unit difference in the input variable. The second difference is the difference between the values of successive first differences found.

Example:

Difference in
x
Input
x
Output
y
First difference
in y
Second difference in y

-9 130
+2
-7 72
-58
+2 -5 30
-42 +16
+2 -3 4 -26
+16
+2 -1 -6 -10
+16
+2
1 0 +6
+16
+2
3 22 +22 +16
+2
5 60 +38
+16
+2
7 114 +54
+16
+2
9 184 +70
+16

It can be seen the second difference in the dependent variable y is a constant = +16 for every increase of 2 units in the independent variable 8.

Hence the the numerical values in the table represents a quadratic function.
The equation to the quadratic function can be assumed as
y = ax2 + bx + c
Substituting the three ordered pairs (-3, 4), (-1, -6) and (1, 0) in the above equation we get a system of equations,

9a - 3b + c = 4 ----------(1)
a - b + c = -6 ----------(2)
a + b + c = 0 ----------(3)

Solving the system we get the values of the arbitrary constants a, b and c as a = 2, b = 3 and c = -5.

Thus the function rule for the table is y = 2x2 + 3x - 5.
The pattern to be looked for identifying an exponential function from the numerical data is a constant ratio between two successive output values when the change in the input values is a constant.

Example:

x

y

-2
0.75
0
3
2
12
4
48
6
192
8
768
10
3072

In the above table, the values of x increases constantly by 2. When we observe the ratio between consecutive y values,

$\frac{3}{0.75}$ = $\frac{12}{3}$ = $\frac{48}{12}$ = $\frac{192}{48}$ = $\frac{768}{192}$ = $\frac{3072}{768}$ = 4

The ratio is a constant 4.

Hence the above table represents an exponential function.

The function can be assumed as y = A0bx. where b the constant ratio of the exponential function is given by the constant ratio in the y values for a unit change in x.

The constant ratio observed in the table is 4 for an increase of 2 units in x values

Thus b = $\frac{4}{2}$ = 2.

The constant A0 in the exponential equation is given by the y value corresponding to x = 0, which is seen as 3 in the table.

Hence the exponential equation for the table is y = 3(2)x.

The constant A0 and the common ratio b can also be determined substituting two ordered pairs in the model equation and solving the system.
A function table can be graphed plotting the input and output values as ordered pairs on Graph.
Generally to graph a function given as an equation, a function table is first made.
The x values are suitably chosen, and the corresponding y values are found using the equation.
While choosing the x values, the following points are to be kept in mind.
  1. The scale to be chosen on the graph.
  2. Smaller numbers are easy to plot.
  3. Irrational and fraction values which do not suit the scale are to be avoided for y.

Example:

let us see how a function table is made for a linear function.
x + 2y = 5.

y = $\frac{-x+5}{2}$ Write the equation solved for y.
The table for the graph can be made as follows:

x
y
-1 3
1 2
3 1

Here the small x values are chosen to get integer values for y. The straight line graph done using the plot is shown here.

Graph Table


Solved Examples

Question 1: Verify whether the numerical values given in the table represent a function. If yes, write a rule for the function table.
  
  x     y  
  -4  16
  -3   9
  -2   4
  -1   1
   0   0
   1
  1
   2
  4
   3
  9
   4
 16


Solution:
 
In the given table, each x value (input) is paired exactly with one y value (output). Hence the table represents a function.

It can be easily observed, that y values are the squares of x values.
Hence the function rule for the table is y = x2.
 

Question 2: Identify the type of relationship for the function table given. Write also the rule describing the function.
    
  x     y 
 -3   -5 
 -1   3
  1  11
  3  19
  5  27

Solution:
 
The linear pattern can be easily recognized here. For every increase of 2 in the x value, y increases constantly by 8.

 Slope of the linear relation m = $\frac{8}{2}$ = 4

 Let the equation to represent the function be y = mx + b

 Substituting the ordered pair (-1, 3) in the equation,

 4(-1) + b = 3      ⇒   b = 7.

 Hence the function rule for the given table is  y = 4x + 7.