this function ( f(x) ), shown in the graph, represents an exponential growth function. compare the rate of…

Explanation:

Step1: Find the average rate of change formula

The average rate of change of a function (y = f(x)) over the interval ([a,b]) is given by (\frac{f(b)-f(a)}{b - a}).

Step2: Calculate the average rate of change for (f(x))

From the graph, when (x = 0), (f(0)=24) and when (x = 4), (f(4)=0). Using the formula (\frac{f(4)-f(0)}{4 - 0}=\frac{0 - 24}{4}=- 6). But since we are talking about magnitude (rate of change), we take the absolute value (\vert - 6\vert=6). Wait, no, re - check. Wait, if we assume the formula for average rate of change (\frac{f(b)-f(a)}{b - a}). If (a = 0), (b = 4). From the graph (assuming (y) - values: at (x = 0), (y=24); at (x = 4), (y = 0)). The average rate of change is (\frac{0 - 24}{4-0}=\frac{-24}{4}=-6). But if we consider the problem might have a wrong initial thought. Wait, another approach: if we assume two points ((x_1,y_1)=(0,24)) and ((x_2,y_2)=(4,0)). The average rate of change (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 24}{4-0}=-6). But maybe the problem is in the calculation of (g(x)). For (g(x)=25(1.4)^x), when (x = 0), (g(0)=25(1.4)^0=25) and when (x = 4), (g(4)=25(1.4)^4). First, calculate ((1.4)^4=(1.4)^2\times(1.4)^2=(1.96)\times(1.96)=3.8416). Then (g(4)=25\times3.8416 = 96.04). The average rate of change of (g(x)) over ([0,4]) is (\frac{g(4)-g(0)}{4 - 0}=\frac{96.04 - 25}{4}=\frac{71.04}{4}=17.76\approx17.8).

Answer:

The average rate of change of (f(x)) is (6) (calculated as (\frac{0 - 24}{4-0}=-6), taking magnitude) and the average rate of change of (g(x)) is approximately (17.8). So (g(x)) has a greater average rate of change.