Answer :
To determine which statement correctly compares the functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] on the interval [tex]\([-1, 2]\)[/tex], let's analyze both functions:
### Function [tex]\( f \)[/tex]
Function [tex]\( f \)[/tex] is represented by the following table of values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & -46 \\
-1 & -22 \\
0 & -10 \\
1 & -4 \\
2 & -1 \\
\hline
\end{array}
\][/tex]
Observing the table, notice that as [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to [tex]\(2\)[/tex], the values of [tex]\( f(x) \)[/tex] are [tex]\(-22, -10, -4, -1\)[/tex]. These values are increasing, which means that [tex]\( f(x) \)[/tex] is increasing on this interval.
Also, note that all values of [tex]\( f(x) \)[/tex] are negative for [tex]\( x = -1, 0, 1, \)[/tex] and [tex]\(2\)[/tex].
### Function [tex]\( g \)[/tex]
Function [tex]\( g \)[/tex] is given by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^x + 2 \][/tex]
We calculate [tex]\( g(x) \)[/tex] for [tex]\( x = -1, 0, 1, \)[/tex] and [tex]\(2\)[/tex]:
- [tex]\( g(-1) = -18 \times 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \times 1 + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
The values for [tex]\( g(x) \)[/tex] are [tex]\(-52, -16, -4, 0\)[/tex], which consistently increase. Thus, [tex]\( g(x) \)[/tex] is also increasing on this interval.
However, not all values of [tex]\( g(x) \)[/tex] are negative; for example, [tex]\( g(2) = 0 \)[/tex].
### Conclusion
From the analysis:
- Function [tex]\( f \)[/tex] is increasing and negative throughout the interval.
- Function [tex]\( g \)[/tex] is increasing, but not all its values are negative.
Thus, the correct statement is:
A. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.
So, the answer is A.
### Function [tex]\( f \)[/tex]
Function [tex]\( f \)[/tex] is represented by the following table of values:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & -46 \\
-1 & -22 \\
0 & -10 \\
1 & -4 \\
2 & -1 \\
\hline
\end{array}
\][/tex]
Observing the table, notice that as [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to [tex]\(2\)[/tex], the values of [tex]\( f(x) \)[/tex] are [tex]\(-22, -10, -4, -1\)[/tex]. These values are increasing, which means that [tex]\( f(x) \)[/tex] is increasing on this interval.
Also, note that all values of [tex]\( f(x) \)[/tex] are negative for [tex]\( x = -1, 0, 1, \)[/tex] and [tex]\(2\)[/tex].
### Function [tex]\( g \)[/tex]
Function [tex]\( g \)[/tex] is given by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^x + 2 \][/tex]
We calculate [tex]\( g(x) \)[/tex] for [tex]\( x = -1, 0, 1, \)[/tex] and [tex]\(2\)[/tex]:
- [tex]\( g(-1) = -18 \times 3 + 2 = -54 + 2 = -52 \)[/tex]
- [tex]\( g(0) = -18 \times 1 + 2 = -18 + 2 = -16 \)[/tex]
- [tex]\( g(1) = -18 \times \frac{1}{3} + 2 = -6 + 2 = -4 \)[/tex]
- [tex]\( g(2) = -18 \times \frac{1}{9} + 2 = -2 + 2 = 0 \)[/tex]
The values for [tex]\( g(x) \)[/tex] are [tex]\(-52, -16, -4, 0\)[/tex], which consistently increase. Thus, [tex]\( g(x) \)[/tex] is also increasing on this interval.
However, not all values of [tex]\( g(x) \)[/tex] are negative; for example, [tex]\( g(2) = 0 \)[/tex].
### Conclusion
From the analysis:
- Function [tex]\( f \)[/tex] is increasing and negative throughout the interval.
- Function [tex]\( g \)[/tex] is increasing, but not all its values are negative.
Thus, the correct statement is:
A. Both functions are increasing, but function [tex]\( g \)[/tex] increases at a faster average rate.
So, the answer is A.