To analyze the functions \( f(x) \) and \( g(x) \), let's look at each one separately.
1. Analyzing \( f(x) = -2x + 5 \)
This is a linear function with a slope of \(-2\). Since the slope is negative, \( f(x) \) is a decreasing function.
2. Analyzing \( g(x) \)
We can evaluate the values from the table to determine if \( g(x) \) is increasing or decreasing. The values in the provided table are:
- \( g(-2) = \frac{1}{4} \)
- \( g(-1) = \frac{1}{2} \)
- \( g(0) = 1 \)
- \( g(1) = 2 \)
- \( g(3) = 8 \)
We can see how \( g(x) \) changes:
- From \( x = -2 \) to \( x = -1 \): \( g(-2) < g(-1) \)
- From \( x = -1 \) to \( x = 0 \): \( g(-1) < g(0) \)
- From \( x = 0 \) to \( x = 1 \): \( g(0) < g(1) \)
- From \( x = 1 \) to \( x = 3 \): \( g(1) < g(3) \)
Since \( g(x) \) is increasing throughout the given values, we conclude that \( g(x) \) is an increasing function.
Conclusion:
- \( f(x) \) is a decreasing function.
- \( g(x) \) is an increasing function.
Thus, the correct answer is:
B: \( g(x) \) is an increasing function, and \( f(x) \) is a decreasing function.