Answer:
A. For every value of x, the rate of change of f exceeds the rate of change of g.
FALSE
Because the rate of change of f increases from 4 to 294, while the rate of change of g is constant (8).
B. As x increases, the rate of change of g exceeds the rate of change of f.
FALSE.
Because the rate of change of g is 8 (constant) and the rate of change of f starts at 4 and increases exceeding that of g.
C. At x = 2, the rate of change of f is equal to the rate of change of g.
FALSE
At x = 2 the rate of change of f is 12 and the rate of change of g is 8.
D. As x increases, the rate of change of f exceeds the rate of change of g.
TRUE
The rate of change of f at start is 4 and increases to 294, while the rate of change of g is 8 (constant), so as x increases, the rate of change of f exceeds the rate of change of g.
Explanation:
The rate of change of a function is calculated as:
- rate of change = rise / run = change in y / change in x = Δy / Δx
For f(x) you get the following rates of change:
x f(x) Δy Δx Δy/Δx
0 3 - - -
1 7 7-3 = 4 1 - 0 = 1 4/1 = 4
2 19 19 - 7 = 12 2 - 1 = 1 12/1 = 12
3 55 55 - 19 = 36 3 - 2 = 1 36
4 163 163 - 55 = 108 4 - 3 = 2 108
5 457 457 - 163 = 294 5 - 4 = 1 294
From that, you see that the function f(x) is an increasing function with an increasing rate of change in the interval [0,5].
For g(x) you get f(x) you get the following rates of change:
x g(x) Δy Δx Δy/Δx
0 3
1 11 11 - 3 = 8 1 - 0 = 1 8
2 19 19 - 11 = 8 2 - 1 = 1 8
3 27 27 - 19 = 8 3 - 2 = 1 8
4 35 35 - 27 = 8 4 - 3 = 1 8
5 43 43 - 35 = 8 5 - 4 = 1 8
From that you can see that the function f(x) is increasing linear function, so its rates of change is constant.
