Question

1. over what intervals is f(x) decreasing?

a) (-1,3) b) (-∞,-1) ∪ (1,∞) c) (-∞,0) d) (-1,1)

Explanation:

Step1: Recall Decreasing Function

A function \( f(x) \) is decreasing on an interval if, as \( x \) increases, \( f(x) \) decreases (i.e., the graph has a negative slope on that interval).

Step2: Analyze the Graph

• From \( x = -1 \) to \( x = 1 \), the graph is going down (slope is negative). Wait, no—wait, let's check the graph again. Wait, the peak is at \( x = -1 \) (since the graph rises to \( x = -1 \), then falls until \( x = 1 \), then rises again). Wait, no, looking at the graph: from \( x = -1 \) to \( x = 1 \), the graph is decreasing? Wait, no, let's parse the options. Wait, the options are A: (-1,3), B: (-∞,-1)∪(1,∞), C: (-∞,0), D: (-1,1). Wait, no—wait, when is the function decreasing? Let's see the graph:

• From \( x = -1 \) to \( x = 1 \), the graph is decreasing (since as \( x \) increases from -1 to 1, \( f(x) \) decreases). Wait, but let's check the options. Wait, option D is (-1,1). Wait, but let's re-examine. Wait, the graph: left part: from \( -\infty \) to \( x = -1 \), it's increasing (since it goes up to \( x = -1 \)). Then from \( x = -1 \) to \( x = 1 \), it's decreasing (goes down to \( x = 1 \)). Then from \( x = 1 \) to \( \infty \), it's increasing (goes up). So the decreasing interval is (-1, 1). Wait, but option D is (-1,1). But wait, let's check the options again. Wait, maybe I misread the graph. Wait, the peak is at \( x = -1 \) (y=3), then it goes down to \( x = 1 \) (y=-1), then up. So from \( x = -1 \) to \( x = 1 \), the function is decreasing. So the interval is (-1,1), which is option D? Wait, no—wait, the options: A is (-1,3), B is (-∞,-1)∪(1,∞), C is (-∞,0), D is (-1,1). Wait, but let's check the slope. When \( x \) is in (-1,1), as \( x \) increases, \( f(x) \) decreases. So the function is decreasing on (-1,1). So the answer is D? Wait, but maybe I made a mistake. Wait, no—wait, let's check the options again. Wait, the problem is "over what intervals is f(x) decreasing". So the correct interval is where the function is decreasing, i.e., where the slope is negative. So from \( x = -1 \) to \( x = 1 \), the slope is negative, so the function is decreasing on (-1,1), which is option D. Wait, but let's confirm. Alternatively, maybe I misread the graph. Wait, the graph: at \( x = -1 \), it's a peak, then it falls until \( x = 1 \) (a valley), then rises. So the decreasing interval is (-1,1), so option D.

Wait, but let's check the options again. Wait, option D is (-1,1). So that's the interval where the function is decreasing.

Answer:

D. (-1,1)