Question

given the absolute function graphed below, find the following:
a) domain:
b) range:
c) on what interval is the function increasing, and on what interval is the function decreasing?
increasing:
decreasing:
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Explanation:

Part a: Find Domain

Step1: Understand Domain Definition

Domain is the set of all x - values for which the function is defined (the x - values covered by the graph).

Step2: Analyze the Graph's x - values

Looking at the graph, the leftmost point has an x - value of - 5, and the graph extends to the right. There are no breaks in the x - values (except for the open circle at (5, - 6), but the domain considers the x - values where the function is defined; the open circle affects the range or a different part, but for domain, we look at all x from - 5 up to 5 (since the graph goes from x=-5 to x = 5, including x=-5 and x = 5? Wait, no, wait. Wait, the graph: the left end is at x=-5, and it goes to x = 5? Wait, the graph has a vertex? Wait, the absolute value graph: let's check the x - values. The graph starts at x=-5 (the leftmost point) and goes to x = 5? Wait, no, looking at the grid, the x - axis is from - 5 to 5. The graph is a V - shaped graph? Wait, no, the blue line: from x=-5, it goes down to a vertex at x=-4 (since at x=-4, the y - value is - 3), then up to x = 5. Wait, no, the domain is all real numbers from the minimum x to maximum x of the graph. Wait, the leftmost x is - 5, and the rightmost x is 5. So the domain is \([-5,5]\)? Wait, no, wait, the open circle is at (5, - 6), but the domain is about x - values. The function is defined for all x from - 5 to 5 (because the graph exists for x from - 5 to 5, even though there's an open circle at (5, - 6), the x - value 5 is still in the domain? Wait, no, the open circle means that the point (5, - 6) is not included, but the domain is the set of x - values. So the x - values go from - 5 to 5, inclusive? Wait, the left end is a closed point? Wait, the leftmost point: at x=-5, the graph has a point (closed? Because it's a solid line there). Then it goes to x = 5, with an open circle at (5, - 6). But for domain, we look at all x where the function is defined. So the x - values range from - 5 to 5, inclusive? Wait, no, the domain is \([-5,5]\) because the graph covers all x from - 5 to 5.

Part b: Find Range

Step1: Understand Range Definition

Range is the set of all y - values (outputs) of the function.

Step2: Analyze the Graph's y - values

The lowest y - value: the vertex is at ( - 4, - 3)? Wait, no, wait, the open circle is at (5, - 6), but is that part of the function? Wait, the blue line: from x=-5, y starts at... Wait, at x=-5, the y - value is... looking at the graph, at x=-5, the y - value is - 1? Wait, no, the grid: each square is 1 unit. Let's re - examine:

- At x=-5, the point is ( - 5, - 1)? Wait, no, the leftmost point: the blue line starts at ( - 5, - 1)? Wait, no, the first point: at x=-5, y=-1? Then it goes down to ( - 4, - 3), then up to (5,5)? Wait, no, the top right point is (5,5). Wait, the open circle is at (5, - 6). So the function's y - values: the minimum y - value: the lowest point on the graph (excluding the open circle, because the open circle is not part of the function). The vertex is at ( - 4, - 3), and then the graph goes up to (5,5). Wait, no, the open circle is at (5, - 6), which is not part of the function. So the function's y - values: from the minimum y (which is - 3, at x=-4) up to the maximum y (which is 5, at x = 5). Wait, no, let's check the graph again. The blue line: from x=-5, it goes down to a minimum at x=-4 (y=-3), then up to x = 5, where y = 5. The open circle is at (5, - 6), which is not part of the function. So the range is all y such that \(y\geq - 3\) and \(y eq - 6\)? Wait, no, the open circle is at (5, - 6), but the main graph (the blue line) has y - values from - 3 up to 5, and also, wait, no, the left part: at x=-5, y=-1; at x=-4, y=-3; then at x=-1, y=-1; at x = 0, y = 0? Wait, maybe I made a mistake. Let's do it properly.

For domain: the domain is the set of all x - coordinates of the points on the graph. The graph starts at x=-5 (the leftmost x) and goes to x = 5 (the rightmost x, even with the open circle at (5, - 6), the x - value 5 is still in the domain because the domain is about where the function is defined, and the open circle is a y - value exclusion, not an x - value exclusion. Wait, no, the open circle means that the point (5, - 6) is not in the function, but the function is defined at x = 5 (since there is a point at (5,5)). Wait, the blue line at x = 5 has a point at (5,5), and an open circle at (5, - 6). So the domain is all x from - 5 to 5, inclusive. So domain is \([-5,5]\).

Part b: Find Range

Step1: Understand Range Definition

Range is the set of all y - values (outputs) of the function.

Step2: Analyze the Graph's y - values

The lowest y - value on the graph (excluding the open circle at (5, - 6), which is not part of the function) is - 3 (at x=-4). The highest y - value is 5 (at x = 5). Also, we need to check if there are any restrictions. The open circle at (5, - 6) means that y=-6 is not in the range, but the main graph has y - values from - 3 up to 5, and also, wait, the left part: at x=-5, y=-1; at x=-4, y=-3; then as x increases from - 4 to 5, y increases from - 3 to 5. Also, is there a lower y? Wait, the vertex is at ( - 4, - 3), so the minimum y - value of the function (the part that is graphed, excluding the open circle) is - 3, and the maximum is 5. Wait, but also, the open circle is at (5, - 6), which is not part of the function, so the range is \(y\geq - 3\) and \(y eq - 6\)? No, that's not right. Wait, the function is the blue line. The blue line has a minimum y - value of - 3 (at x=-4) and a maximum y - value of 5 (at x = 5). The open circle is a separate point that is not part of the function, so the range is all real numbers from - 3 up to 5, inclusive? Wait, no, let's check the y - values:

- At x=-5, y=-1 (since the point is ( - 5, - 1))
- At x=-4, y=-3 (vertex)
- At x=-1, y=-1
- At x = 0, y = 0
- At x = 3, y = 3
- At x = 5, y = 5

So the y - values start at - 1 (x=-5), go down to - 3 (x=-4), then up to 5 (x = 5). Wait, so the minimum y - value is - 3, and the maximum is 5. So the range is \([-3,5]\)? Wait, but the open circle is at (5, - 6), which is not part of the function, so that y - value (-6) is not in the range, but our function (the blue line) has y - values from - 3 to 5. So range is \([-3,5]\).

Part c: Increasing and Decreasing Intervals

Step1: Understand Increasing/Decreasing

A function is increasing on an interval if, as x increases, y increases. A function is decreasing on an interval if, as x increases, y decreases.

Step2: Find the Vertex x - value

The vertex of the absolute - value - like graph is at x=-4 (since that's where the graph changes from decreasing to increasing).

Step3: Determine Decreasing Interval

To the left of the vertex (x < - 4), as x increases (from - 5 to - 4), y decreases (from - 1 to - 3). So the function is decreasing on the interval \([-5, - 4]\).

Step4: Determine Increasing Interval

To the right of the vertex (x > - 4), as x increases (from - 4 to 5), y increases (from - 3 to 5). So the function is increasing on the interval \([-4,5]\).

Answer:

a) Domain: \(\boldsymbol{[-5, 5]}\)
b) Range: \(\boldsymbol{[-3, 5]}\)
c) Increasing: \(\boldsymbol{[-4, 5]}\), Decreasing: \(\boldsymbol{[-5, - 4]}\)