The process of determining the linearity of a function involves evaluating the constant rate of change (slope) between points. For the given function with points (1,5), (2,10), (3,7), and (4,14), the slopes differ, indicating a non-linear function. The value of 'n' influencing linearity is indeterminable with the provided information.
To determine the value of n that makes f a linear function, we need to look at the relationship between the x values and the corresponding f(x) values. For a function to be linear, the rate of change between any two points on the function should be constant. This means that the difference in the y-values divided by the difference in the x-values (also known as the slope) should be the same for all pairs of points.
Given the points (1,5), (2,10), (3,7), and (4,14), we can calculate the slopes between consecutive points:
The slope between points (1,5) and (2,10) is (10 - 5) / (2 - 1) = 5/1 = 5.
The slope between points (2,10) and (3,7) is (7 - 10) / (3 - 2) = -3/1 = -3.
The slope between points (3,7) and (4,14) is (14 - 7) / (4 - 3) = 7/1 = 7.
Since the slopes are not the same, the given points do not form a linear function as is. Nowhere in the provided information is there enough detail to solve for n directly that would make this function linear; more information about n or additional points would be necessary to determine such a value.