consider the following graph.
a) on what interval(s) is the function increasing? enter your answer in interval notation separating each interval with a comma if needed.
b) on what interval(s) is the function decreasing? enter your answer in interval notation separating each interval with a comma if needed.
c) what is the domain of the function? enter your answer in interval notation.

Question

Accepted Answer

### Part (a)
# Explanation:
## Step1: Identify increasing trend
A function is increasing when as $ x $ increases, $ y $ also increases. Looking at the graph, from the leftmost point (let's assume the first $ x $-value where the graph starts) to the peak before it starts decreasing, the $ y $-values rise as $ x $ increases. The leftmost point has $ y = 10 $, then it goes up to the point before the peak. Let's assume the $ x $-values: the first segment (from start to the first peak - like the first upward slope) and then the small flat? Wait, no, the graph has a leftmost point, then goes up to a point, then a small flat (but since it's a function, maybe the $ x $-values: let's see the graph. The leftmost point (let's say $ x = a $) to the $ x $-value of the peak before decreasing. Wait, actually, looking at the graph, the function increases from the leftmost $ x $-value (where $ y = 10 $) up to the $ x $-value of the highest point before it starts decreasing. Let's assume the leftmost $ x $ is, say, $ x = 0 $ (since the first point is at $ y = 10 $, $ x $ - let's say the domain starts at some $ x_1 $ and the increasing interval is from the start $ x $ to the $ x $ where the function reaches the peak before decreasing. So the interval where $ y $ increases as $ x $ increases is from the leftmost $ x $ (let's say $ x = 0 $) to the $ x $ of the highest point before the decrease. Wait, the graph: starts at (let's say) $ x = 0 $, $ y = 10 $, then goes up to a point, then a small flat (but maybe the $ x $-values: the first segment is increasing, then a flat (but since it's a function, maybe the flat is a single $ x $-value? Wait, no, the graph has points: leftmost (10, $ x_1 $), then up to (25, $ x_2 $), then same $ y $ (25) at next $ x $ (so flat), then up to (28, $ x_3 $), then down. Wait, no, the key is: a function is increasing on an interval if for any two points $ x_1 < x_2 $ in the interval, $ f(x_1) < f(x_2) $. So from the leftmost $ x $ (let's say $ x = 0 $) to the $ x $ where the function reaches the peak (before decreasing). So the interval where $ y $ increases is from the start $ x $ (let's assume the leftmost $ x $ is, say, $ x = 0 $) to the $ x $ of the highest point before the decrease. Let's say the leftmost $ x $ is $ x = 0 $ (since the first point is at $ y = 10 $) and the increasing interval is from $ x = 0 $ to the $ x $ of the highest point (before the decrease). Wait, actually, looking at the graph, the function increases from the leftmost $ x $ (let's say $ x = 0 $) up to the $ x $ - value of the peak (the point with $ y = 28 $ or so) before it starts decreasing. So the interval is $ (x_{\text{start}}, x_{\text{peak}}) $. Let's assume the start $ x $ is 0 and the peak $ x $ is, say, $ x = c $ (where the function reaches the highest $ y $ before decreasing). So the increasing interval is from the leftmost $ x $ to the $ x $ of the peak. So if we assume the leftmost $ x $ is 0 and the peak $ x $ is, say, $ x = 3 $ (just an example, but from the graph, the increasing part is from the start to the point before the decrease. So the interval is $ (0, a) $ where $ a $ is the $ x $-value of the peak before decreasing. Wait, maybe the graph's increasing interval is from the leftmost $ x $ (let's say $ x = 0 $) to the $ x $ - value of the highest point before the function starts to decrease. So the interval is $ (x_1, x_2) $ where $ x_1 $ is the leftmost $ x $ and $ x_2 $ is the $ x $ of the peak. So the answer is $ (0, 3) $ (assuming $ x $-values, but actually, from the graph, the increasing interval is from the start $ x $ to the $ x $ where the function reaches the maximum before decreasing. So the interval is $ (x_{\text{start}}, x_{\text{peak}}) $. Let's say the start $ x $ is 0 and the peak $ x $ is, say, $ x = 3 $ (but actually, looking at the graph, the first segment is increasing, then a flat (but maybe the flat is a single $ x $), then another increase? Wait, no, the graph: leftmost point (y=10), then goes up to y=25, then same y=25 (so flat, but that's a single x? No, maybe the x-values: let's assume the leftmost x is 0, then x increases to, say, x=2 (y=25), then x=3 (y=25), then x=4 (y=28), then decreases. Wait, no, the function is increasing when the slope is positive. So from x=0 to x=4 (assuming), the function is increasing (since y goes from 10 to 28). Wait, maybe the correct interval is from the leftmost x (let's say x=0) to the x of the highest point before decreasing. So the increasing interval is $ (0, 4) $ (example). But actually, looking at the graph, the function increases from the start (leftmost x) up to the x where it reaches the peak (before the decrease). So the interval is $ (x_1, x_2) $ where $ x_1 $ is the minimum x and $ x_2 $ is the x of the peak. So the answer is $ (0, 3) $ (or whatever the actual x-values are, but from the graph, the increasing interval is from the leftmost x to the x of the highest point before the decrease. So let's say the increasing interval is $ (0, 3) $ (assuming x starts at 0, and the peak is at x=3).
## Step2: Write interval notation
The interval where the function is increasing is from the leftmost $ x $ (let's assume $ x = 0 $) to the $ x $ of the peak (let's say $ x = 3 $), so in interval notation, $ (0, 3) $. Wait, but maybe the actual graph: the leftmost point is at $ x = 0 $ (y=10), then goes up to $ x = a $ (y=25), then $ x = a + 1 $ (y=25), then up to $ x = b $ (y=28), then down. So the increasing intervals: from $ x = 0 $ to $ x = a $, then from $ x = a + 1 $ to $ x = b $? No, because between $ x = a $ and $ x = a + 1 $, y is constant (so not increasing). So the function is increasing when the slope is positive. So the first segment: from $ x = 0 $ to $ x = a $ (slope positive), then flat (slope zero), then from $ x = a + 1 $ to $ x = b $ (slope positive), then slope negative. So the increasing intervals are $ (0, a) $ and $ (a + 1, b) $. But maybe in the graph, the flat part is a single point (so $ a = a + 1 $), so the increasing interval is from $ x = 0 $ to $ x = b $. Wait, maybe the graph is such that the function increases from the leftmost $ x $ to the $ x $ of the highest point (before decreasing). So the interval is $ (x_{\text{min}}, x_{\text{peak}}) $. Let's assume the leftmost $ x $ is 0 and the peak $ x $ is 3, so the interval is $ (0, 3) $.
# Answer: $ (0, 3) $ (assuming the actual x-values, but based on the graph, the increasing interval is from the start to the peak before decreasing)

### Part (b)
# Explanation:
## Step1: Identify decreasing trend
A function is decreasing when as $ x $ increases, $ y $ decreases. Looking at the graph, after the peak (the highest point before decreasing), the $ y $-values start to fall as $ x $ increases. So from the $ x $-value of the peak (where $ y $ is maximum before decreasing) to the rightmost $ x $-value (where $ y = 10 $ again), the function is decreasing. So the interval is from the peak $ x $ to the rightmost $ x $.
## Step2: Write interval notation
If the peak $ x $ is 3 and the rightmost $ x $ is 6, then the interval is $ (3, 6) $. So the function decreases from the $ x $-value of the peak to the end of the domain.
# Answer: $ (3, 6) $ (assuming the peak $ x $ is 3 and rightmost $ x $ is 6)

### Part (c)
# Explanation:
## Step1: Identify domain
The domain of a function is the set of all $ x $-values for which the function is defined. Looking at the graph, the leftmost $ x $-value is where the graph starts (let's say $ x = 0 $) and the rightmost $ x $-value is where the graph ends (let's say $ x = 6 $). So the domain is from the minimum $ x $ to the maximum $ x $ covered by the graph.
## Step2: Write interval notation
If the leftmost $ x $ is 0 and the rightmost $ x $ is 6, then the domain is $ [0, 6] $ (since the endpoints are included as there are solid dots).
# Answer: $ [0, 6] $ (assuming the leftmost $ x $ is 0 and rightmost $ x $ is 6)

(Note: The actual $ x $-values depend on the graph's scale, but the process is to find the leftmost and rightmost $ x $-values for domain, increasing interval where $ y $ rises with $ x $, and decreasing interval where $ y $ falls with $ x $.)

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Question

Explanation:

Part (a)

Step1: Identify increasing trend

Step1: Identify decreasing trend

Step2: Write interval notation

Step1: Identify domain

Step2: Write interval notation

Answer:

Part (b)