Question

Your mathematics teacher asks you to sketch a graph of the exponential function f(x) = (3 / 2)^x for x, a number

between 0 and 40 inclusively, using a scale of 10 units to one inch for both the x- and y-axes.
c. Find an m so that the linear function ????(x) = mx + 2 is greater than f(x) for all x such that 0 ≤ x ≤ 40, but
f(41) > ????(41)

Answer 1

I think there is a mistake in the question. If f(x) = (3 / 2)^x

, then

x f(x)

0 1

40 11057332.32

If the scale is 10 units to one inch, then you will need a paper size of 1105733.232 inches!

As a reference, a graph is included in attachments.

To get a linear function 2 points are needed. Assuming the equation g(x) = m*x and taking the points of the aforementioned table, we can compute the slope as

m = (f(x2) - f(x1))/(x2 - x1)

m = (11057332.32 - 1)/(40 - 0)

m = 276434.283

Function g(x) = 276434.283*x pass at points (0, 1) and (40, 11057332.32). h(x) = 276434.283*x + 2 fulfill the requirement that h(x) is greater than f(x), for all x such that 0 ≤ x ≤ 40, but f(41) > h(41), because at the next point, the exponential function will growths faster than linear function (see figure).