Determine whether the function given by the table is linear

Determine whether the function given by the table is linear, exponential, or neither. If the function is linear, find a linear function that models the data; if it is exponential, find an exponential function that models the data. x (-1, 0, 1, 2, 3) and f(x) (0, 4, 8, 12, 16)

Solution

Let f(x) = y. We have been fiven the y-values (0, 4, 8, 12, 16) corresponding to the x -values (-1, 0, 1, 2, 3). If if plot these points in the cartesian plane, we get a straight line as the graph of y = f(x). Therefore, f(x) is a linear function. Let the equation y = mx + c describe the equation of the linear function. Then on substituting x = -1 and y = 0 in this equation, we get - m + c = 0...(1). Further, on substituting x = 0 and y = 4 in this equation, we get 0 + c = 4 so that c = 4. Now, on substituting c = 4 in equation number 1, we get - m + 4 = 0 so that m = 4. Then the required linear function is f (x) = y = 4x + 4. We can verfy the result by substituting x = 1,2,and 3 in this equation. On these substitutions, we get the corresponding values of f (x) as 8, 12 and 16. Thus, the required linear function is f (x) = y = 4x + 4.