Answer :
To solve this problem, we need to compare the two functions, [tex]\( f \)[/tex] and [tex]\( g \)[/tex], over the interval [tex]\([-1, 2]\)[/tex] and see which statements about the functions are true. Here's how we can do it step by step:
### Function [tex]\( f \)[/tex]
1. Values for Function [tex]\( f \)[/tex]: Looking at the given table for function [tex]\( f \)[/tex], we see:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
2. Determine If [tex]\( f \)[/tex] Is Increasing: We check if each consecutive value of [tex]\( f(x) \)[/tex] is greater than the previous value, which would indicate an increasing function:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\(-10 > -22\)[/tex]
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\(-4 > -10\)[/tex]
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\(-1 > -4\)[/tex]
All comparisons show that [tex]\( f(x) \)[/tex] values are increasing.
3. Check If [tex]\( f(x) \)[/tex] Is Negative on [tex]\([-1, 2]\)[/tex]: All values of [tex]\( f(x) \)[/tex] we listed ([tex]\(-22, -10, -4, -1\)[/tex]) are negative in the given interval.
### Function [tex]\( g \)[/tex]
1. Equation for Function [tex]\( g \)[/tex]: The function [tex]\( g \)[/tex] is given by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^3 + 2 \][/tex]
Notice that the equation does not depend on [tex]\( x \)[/tex]. Instead, it simplifies to:
[tex]\[ g(x) = -18 \times \left(\frac{1}{27}\right) + 2 \][/tex]
[tex]\[ g(x) = -\frac{18}{27} + 2 \][/tex]
[tex]\[ g(x) = -\frac{2}{3} + 2 \][/tex]
[tex]\[ g(x) = \frac{4}{3} \][/tex]
2. Value of [tex]\( g(x) \)[/tex]: Since [tex]\( g(x) \)[/tex] evaluates to a constant number ([tex]\(\frac{4}{3}\)[/tex]), the function [tex]\( g(x) \)[/tex] doesn’t change with [tex]\( x \)[/tex]. This means it is not increasing or decreasing; it stays constant.
### Comparison Over the Interval [tex]\([-1, 2]\)[/tex]
- Function [tex]\( f \)[/tex] is increasing since its values increase as [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to [tex]\( 2\)[/tex].
- Function [tex]\( g \)[/tex] is not increasing because it is constant.
- Function [tex]\( f \)[/tex] is negative over the interval, as shown above.
### Conclusion
Based on these observations, the correct statement is:
- Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.
Therefore, the answer is:
D. Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.
### Function [tex]\( f \)[/tex]
1. Values for Function [tex]\( f \)[/tex]: Looking at the given table for function [tex]\( f \)[/tex], we see:
- [tex]\( f(-1) = -22 \)[/tex]
- [tex]\( f(0) = -10 \)[/tex]
- [tex]\( f(1) = -4 \)[/tex]
- [tex]\( f(2) = -1 \)[/tex]
2. Determine If [tex]\( f \)[/tex] Is Increasing: We check if each consecutive value of [tex]\( f(x) \)[/tex] is greater than the previous value, which would indicate an increasing function:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]: [tex]\(-10 > -22\)[/tex]
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]: [tex]\(-4 > -10\)[/tex]
- From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]: [tex]\(-1 > -4\)[/tex]
All comparisons show that [tex]\( f(x) \)[/tex] values are increasing.
3. Check If [tex]\( f(x) \)[/tex] Is Negative on [tex]\([-1, 2]\)[/tex]: All values of [tex]\( f(x) \)[/tex] we listed ([tex]\(-22, -10, -4, -1\)[/tex]) are negative in the given interval.
### Function [tex]\( g \)[/tex]
1. Equation for Function [tex]\( g \)[/tex]: The function [tex]\( g \)[/tex] is given by the equation:
[tex]\[ g(x) = -18\left(\frac{1}{3}\right)^3 + 2 \][/tex]
Notice that the equation does not depend on [tex]\( x \)[/tex]. Instead, it simplifies to:
[tex]\[ g(x) = -18 \times \left(\frac{1}{27}\right) + 2 \][/tex]
[tex]\[ g(x) = -\frac{18}{27} + 2 \][/tex]
[tex]\[ g(x) = -\frac{2}{3} + 2 \][/tex]
[tex]\[ g(x) = \frac{4}{3} \][/tex]
2. Value of [tex]\( g(x) \)[/tex]: Since [tex]\( g(x) \)[/tex] evaluates to a constant number ([tex]\(\frac{4}{3}\)[/tex]), the function [tex]\( g(x) \)[/tex] doesn’t change with [tex]\( x \)[/tex]. This means it is not increasing or decreasing; it stays constant.
### Comparison Over the Interval [tex]\([-1, 2]\)[/tex]
- Function [tex]\( f \)[/tex] is increasing since its values increase as [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to [tex]\( 2\)[/tex].
- Function [tex]\( g \)[/tex] is not increasing because it is constant.
- Function [tex]\( f \)[/tex] is negative over the interval, as shown above.
### Conclusion
Based on these observations, the correct statement is:
- Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.
Therefore, the answer is:
D. Only function [tex]\( f \)[/tex] is increasing, and only function [tex]\( f \)[/tex] is negative.