Answer:
Step-by-step explanation:
The question is asking you to identify the domain for each of the parts of the piecewise-defined function. This means you need to identify the corresponding part on the graph, and determine its horizontal extent.
<h3>Observation</h3>
The first step in solving a problem is to look at the given information. The graph is shown as three (3) non-overlapping horizontal line segments. The open- or filled-circle end points tell you whether that x-value is part of the line segment.
The blanks in the function definition are where the <em>domain descriptions</em> go for the piecewise-defined function. This means you have to match the function description to the left of "if x ∈" with the set of x-values it corresponds to. The "x ∈" notation suggests you want to specify the domain in "interval notation," rather than as an inequality.
<h3>Bottom piece</h3>
The first part of the function definition is ...
f(x) = -4
This matches the horizontal line segment at the bottom of the graph. The closed dots at its ends mean it is defined for ...
-6 ≤ x ≤ -2
In interval notation, a square bracket is used when the end point is included in the interval.
f(x) = -4 if x ∈ [-6, -2]
<h3>Middle piece</h3>
The second part of the function definition is ...
f(x) = -3
This matches the horizontal line segment in the middle of the graph. The open dot at the left end, and the closed dot at the right end mean it is defined for ...
-2 < x ≤ 1
We use a round bracket (parenthesis) when an end point is not included in the interval.
f(x) = -3 if x ∈ (-2, 1]
<h3>Top piece</h3>
The third part of the function definition is ...
f(x) = 4
This matches the horizontal line segment at the top of the graph. The open dot at the left end, and the closed dot at the right end mean it is defined for ...
1 < x ≤ 5
In interval notation, this is ...
f(x) = 4 if x ∈ (1, 5]
