You are asking how to determine a linear function from a table and a graph.

Linear functions graph as a straight line, no curves allowed. So, if the graph is a straight line, it is the graph of a linear function.

From a table, you can verify a linear function by examining the x and y values.  The rate of change for y with respect to x remains constant for a linear function. This rate of change is called the slope.

We'll use this table for the example.

x        y

0        2

1        3

2        4

3        5

4        6

In this example the rate of change between the x and y values is always 2.  This function could be written with the linear equation y = x + 2.

A linear function graphs as a straight line.

A table of values for a linear function shows a constant rate of change between the x and y values.

A linear function is one that has the form f(x) = ax + b. Here for each value of x there is only one corresponding value of f(x) and every value of f(x) is due to only one particular value of x. In other words there is a one to one correspondence between values of f(x) and values of x.

If the values of f(x) and x for a linear function are written in the form of a table we find that each value of f(x) has a corresponding value of x that is unique. If a graph is drawn of the function it is always a straight line.

A linear function results in a graph that is a straight line, and the values of f(x) and x written as a table show a one to one correspondence.