Question

all points of the step function f(x) are graphed. what is the domain of f(x)? options: {x| -4 < x ≤ 4}, {x| -3 < x ≤ 4}, {x| 1 < x ≤ 4}, {x| 2 < x ≤ 4}

Explanation:

Step1: Identify the leftmost and rightmost x - values

The leftmost point (open circle) is at \(x=-4\), so the domain starts with \(x > - 4\)? Wait, no, looking at the graph: the first segment has an open circle at \(x = - 4\) and a closed circle at \(x=-2\). The next segment: open at \(x=-2\), closed at \(x = 0\). Then open at \(x = 0\), closed at \(x = 2\). Then open at \(x = 2\), closed at \(x = 4\). Wait, no, let's check the x - coordinates of the open and closed circles. The leftmost open circle is at \(x=-4\) (since the first segment starts with an open circle at \(x=-4\)) and the rightmost closed circle is at \(x = 4\). Wait, no, let's list the intervals:

First segment: open at \(x=-4\), closed at \(x=-2\) (so \( - 4

Second segment: open at \(x=-2\), closed at \(x = 0\) (so \( - 2

Third segment: open at \(x = 0\), closed at \(x = 2\) (so \(0

Fourth segment: open at \(x = 2\), closed at \(x = 4\) (so \(2

Wait, but when we take the union of all these intervals, the overall domain's left - most limit: the smallest \(x\) for which the function is defined. The leftmost open circle is at \(x=-4\), and as we move right, the intervals cover from \(x>-4\) up to \(x = 4\)? Wait, no, looking at the options, the first option is \(\{x|-4 < x\leq4\}\). Let's check the rightmost point: the last closed circle is at \(x = 4\), and the leftmost open circle is at \(x=-4\). So the domain is all real numbers \(x\) such that \(-4

Wait, let's check the options again. The options are:

1. \(\{x|-4 < x\leq4\}\)

1. \(\{x|-3 < x\leq4\}\)

1. \(\{x|1 < x\leq4\}\)

1. \(\{x|2 < x\leq4\}\)

Looking at the graph, the leftmost open circle is at \(x=-4\) (since the first segment starts with an open circle at \(x=-4\)) and the rightmost closed circle is at \(x = 4\). So the domain is \(x\) where \(-4 < x\leq4\).

Step2: Match with the options

The first option is \(\{x|-4 < x\leq4\}\), which matches our analysis of the leftmost (open at \(-4\)) and rightmost (closed at \(4\)) x - values of the function's graph.

Answer:

\(\{x|-4 < x\leq4\}\) (the first option in the list of choices)