To determine the correct answer, let's find the average rate of change for both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] over the interval [tex]\([-3, 0]\)[/tex].
### Average Rate of Change of [tex]\( f \)[/tex]
1. Identify the values of [tex]\( f(x) \)[/tex] at [tex]\( x = -3 \)[/tex] and [tex]\( x = 0 \)[/tex]:
[tex]\[
f(-3) = -4.5
\][/tex]
[tex]\[
f(0) = 0
\][/tex]
2. Use the formula for the average rate of change:
[tex]\[
\text{average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Where [tex]\( x_1 = -3 \)[/tex] and [tex]\( x_2 = 0 \)[/tex]:
[tex]\[
\text{average rate of change of } f = \frac{f(0) - f(-3)}{0 - (-3)} = \frac{0 - (-4.5)}{0 + 3} = \frac{4.5}{3} = 1.5
\][/tex]
### Average Rate of Change of [tex]\( g \)[/tex]
1. Identify the given points [tex]\( (-3, 5) \)[/tex] and [tex]\( (0, 14) \)[/tex].
2. Use the same average rate of change formula for these points:
[tex]\[
\text{average rate of change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}
\][/tex]
Where [tex]\( x_1 = -3 \)[/tex] and [tex]\( x_2 = 0 \)[/tex]:
[tex]\[
\text{average rate of change of } g = \frac{g(0) - g(-3)}{0 - (-3)} = \frac{14 - 5}{0 + 3} = \frac{9}{3} = 3
\][/tex]
### Comparison
- Average rate of change of [tex]\( f \)[/tex] over [tex]\([-3, 0]\)[/tex] is [tex]\( 1.5 \)[/tex].
- Average rate of change of [tex]\( g \)[/tex] over [tex]\([-3, 0]\)[/tex] is [tex]\( 3 \)[/tex].
By comparing these rates:
- [tex]\( 1.5 < 3 \)[/tex]
Thus, the correct statement is:
A. The average rate of change of [tex]\( f \)[/tex] is less than the average rate of change of [tex]\( g \)[/tex].
