Question

question
determine all intervals on which the graph of ( f ) is decreasing.
graph of ( f ) (coordinate plane with a function curve)
answer (interval notation)
input box and interval/number buttons, \all real numbers\ option

Explanation:

Step1: Recall decreasing function definition

A function \( f(x) \) is decreasing on an interval if, as \( x \) increases, \( f(x) \) decreases (i.e., the graph falls from left to right).

Step2: Analyze the graph

Looking at the graph:
- From \( x = -\infty \) to \( x = -2 \) (or the leftmost part to \( x=-2 \)), the graph is decreasing (falls as \( x \) increases).
- Then, from \( x = -2 \) to \( x = 1 \) (or some point around \( x = 1 \)), wait, no—wait, let's re - examine. Wait, the left part: starting from the left, the graph goes down until a minimum, then? Wait, actually, looking at the graph (from the description: left side has a line decreasing, then a curve that has a minimum, then increases, then another increase). Wait, the key intervals:
- First, the left - most segment: as \( x \) increases from \( -\infty \) to \( -2 \) (assuming the vertex or the point where it starts to change is at \( x=-2 \)), the function is decreasing.
- Then, from \( x=-2 \) to \( x = 1 \) (or the point where the curve starts to increase more steeply), wait, no—wait, the function decreases from \( -\infty \) to \( -2 \), then from \( -2 \) to \( 1 \) (maybe? Wait, actually, the standard way: a function is decreasing when the slope of the tangent (or the direction of the graph) is downward as \( x \) increases.
- Wait, looking at the graph: the left part (before \( x=-2 \)): as \( x \) increases, \( y \) decreases. Then, from \( x=-2 \) to \( x = 1 \) (let's say the minimum is at \( x=-2 \) and then it goes up? No, wait, maybe the graph: the left line is decreasing, then a curve that has a minimum at some \( x=-2 \), then increases, then after a point (maybe \( x = 3 \)) it increases more. Wait, maybe the correct intervals: the function is decreasing on \( (-\infty, - 2)\) and wait, no—wait, maybe I misread. Wait, the problem is to find where the graph is decreasing. Let's think again. A function is decreasing when, as \( x \) increases, \( f(x) \) decreases. So, looking at the graph:
- From \( x=-\infty \) to \( x=-2 \): the graph is going down (decreasing).
- Then, from \( x=-2 \) to \( x = 1 \): wait, no, maybe the graph has a minimum at \( x=-2 \), then increases until \( x = 3 \), then increases more? Wait, maybe the correct interval is \( (-\infty, - 2)\) and wait, no—wait, maybe the graph is decreasing on \( (-\infty, - 2)\) and also, wait, no, let's check the x - axis. The left part: when \( x \) increases from, say, \( -6 \) to \( -2 \), \( y \) decreases. Then, from \( -2 \) to \( 1 \), \( y \) increases (since it goes from the minimum up). Then, after \( x = 3 \), it increases more. Wait, maybe the only decreasing interval is \( (-\infty, - 2)\)? No, wait, maybe I made a mistake. Wait, the graph: the left line is decreasing, then a curve that dips down? Wait, no, the original graph (from the user's image): the left side has a line that is decreasing (from left to right, going down), then a curve that has a minimum (so it decreases until the minimum, then increases). Wait, maybe the x - values: let's assume the minimum is at \( x=-2 \), so from \( -\infty \) to \( -2 \), the function is decreasing, and then from \( -2 \) to \( 1 \) (or some point) it's increasing, then from \( 1 \) to \( 3 \) maybe increasing, then more. Wait, but the standard way: to find where \( f(x) \) is decreasing, we look at the intervals where the graph is falling as \( x \) increases. So, if the graph falls from \( -\infty \) to \( x=-2 \), then that's the decreasing interval.

Wait, maybe the correct interval is \( (-\infty, - 2) \). Wait, but let's confirm. Let's take two points: for \( x_1 < x_2 < - 2 \), \( f(x_1)>f(x_2) \), so it's decreasing. Then, for \( x_1 < x_2 \) where \( x_1>-2 \) and \( x_2>-2 \), \( f(x_1)<f(x_2) \) (increasing). So the function is decreasing on \( (-\infty, - 2) \).

Answer:

\((-\infty, - 2)\)