x g(x)

Question

x	g(x)
−2	[[1/4]]
−1	[[1/2]]
0	1
1	2
3	8

The function g(x) is shown on the table. Suppose f(x) = −2x + 5. Which statement is true when comparing the two functions?
Responses
A	f(x) is an increasing function, and g(x) is a decreasing function.f(x) is an increasing function, and g(x) is a decreasing function.
B	g(x) is an increasing function, and f(x) is a decreasing function.g(x) is an increasing function, and f(x) is a decreasing function.
C	Both functions are increasing.Both functions are increasing.
D	Both functions are decreasing.

GPT-4o mini · Answer

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.