Consider the table of values for functions f and g below.

x f(x) g(x)
0 3 3
1 7 11
2 19 19
3 55 27
4 163 35
5 457 43

which of the following statements is true?

a.
for every value of x, the rate of change of f exceeds the rate of change of g.
b.
as x increases, the rate of change of g exceeds the rate of change of f.
c.
at x = 2, the rate of change of f is equal to the rate of change of g.
d.
as x increases, the rate of change of f exceeds the rate of change of g.

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:

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.