To determine the order of the functions [tex]\( f \)[/tex], [tex]\( g \)[/tex], and [tex]\( h \)[/tex] based on their average rate of change over the interval [tex]\([0, 3)\)[/tex], we need to perform the following steps:
### Step 1: Calculate the average rate of change for [tex]\( f(x) \)[/tex]
Given the values of [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & -1 & 0 & 1 & 2 & 3 & 4 \\
\hline
f(x) & -2 & 1 & 4 & 7 & 10 & 13 \\
\hline
\end{array}
\][/tex]
We focus on the interval [tex]\([0, 3)\)[/tex]:
- [tex]\( f(0) = 1 \)[/tex]
- [tex]\( f(3) = 10 \)[/tex]
The average rate of change of [tex]\( f \)[/tex] over [tex]\([0,3)\)[/tex] is given by:
[tex]\[
\frac{f(3) - f(0)}{3 - 0} = \frac{10 - 1}{3 - 0} = \frac{9}{3} = 3.0
\][/tex]
### Step 2: Calculate the average rate of change for [tex]\( h(x) \)[/tex]
The function [tex]\( h(x) \)[/tex] is defined as:
[tex]\[
h(x) = x^2 + x - 6
\][/tex]
We evaluate [tex]\( h(x) \)[/tex] at the boundaries of the interval [tex]\([0, 3)\)[/tex]:
- [tex]\( h(0) = 0^2 + 0 - 6 = -6 \)[/tex]
- [tex]\( h(3) = 3^2 + 3 - 6 = 9 + 3 - 6 = 6 \)[/tex]
The average rate of change of [tex]\( h \)[/tex] over [tex]\([0,3)\)[/tex] is given by:
[tex]\[
\frac{h(3) - h(0)}{3 - 0} = \frac{6 - (-6)}{3 - 0} = \frac{6 + 6}{3} = \frac{12}{3} = 4.0
\][/tex]
### Step 3: Calculate the average rate of change for [tex]\( g(x) \)[/tex]
Assume the function [tex]\( g(x) \)[/tex] is defined as:
[tex]\[
g(x) = x^2 + 3x - 4
\][/tex]
We evaluate [tex]\( g(x) \)[/tex] at the boundaries of the interval [tex]\([0, 3)\)[/tex]:
- [tex]\( g(0) = 0^2 + 3(0) - 4 = -4 \)[/tex]
- [tex]\( g(3) = 3^2 + 3(3) - 4 = 9 + 9 - 4 = 14 \)[/tex]
The average rate of change of [tex]\( g \)[/tex] over [tex]\([0,3)\)[/tex] is given by:
[tex]\[
\frac{g(3) - g(0)}{3 - 0} = \frac{14 - (-4)}{3 - 0} = \frac{14 + 4}{3} = \frac{18}{3} = 6.0
\][/tex]
### Step 4: Order the functions by their average rate of change
We have the average rates of change for each function:
- [tex]\( f \)[/tex]: 3.0
- [tex]\( h \)[/tex]: 4.0
- [tex]\( g \)[/tex]: 6.0
Ordering these rates from least to greatest, we get:
1. [tex]\( f \)[/tex] (3.0)
2. [tex]\( h \)[/tex] (4.0)
3. [tex]\( g \)[/tex] (6.0)
Thus, the order of the functions from least to greatest according to their average rate of change over the interval [tex]\([0, 3)\)[/tex] is:
[tex]\[
\boxed{f, h, g}
\][/tex]
