Question

the step function f(x) is graphed. what is the value of f(-1)? -3 -1 0 1

Explanation:

Step1: Locate x = -1 on the graph

Find the point on the x - axis corresponding to \(x=-1\).

Step2: Determine the y - value at x = -1

Look at the step function graph. At \(x = - 1\), the filled dot (indicating the value of the function at that point) is at \(y=-1\). So we check the y - coordinate of the point where \(x=-1\) on the step function. The graph shows that for \(x=-1\), the function value \(f(-1)\) is the y - value of the filled circle at \(x = - 1\), which is \(-1\)? Wait, no, wait. Wait, looking at the graph: the left - most segment has a filled dot at \(x=-4\) (y=-3) and open at \(x=-2\). Then the next segment: filled at \(x=-2\)? Wait, no, the graph: at \(x=-1\), the filled dot is at \(y = - 1\)? Wait, no, let's re - examine. The x - axis: the points. Let's see the segments:

First segment: from \(x=-4\) (filled, y=-3) to \(x=-2\) (open). Then next segment: from \(x=-2\) (filled? Wait, no, the dot at \(x=-1\) is filled. Wait, the graph: at \(x=-1\), the filled circle is at \(y=-1\)? Wait, no, the y - axis: the first segment (left) has y=-3, then the next segment (from \(x=-2\) to \(x = 1\) maybe?) has y=-1? Wait, the dot at \(x=-1\) is filled, and its y - coordinate is - 1? Wait, no, looking at the graph again: the vertical line at \(x=-1\), the filled dot is at \(y=-1\)? Wait, no, the y - axis labels: 4,3,2,1,0,-1,-2,-3,-4. The dot at \(x=-1\) is at y=-1? Wait, no, the user's graph: let's parse the graph.

Looking at the x - axis: the points. The first (left - most) horizontal segment: starts at \(x=-4\) (filled, y=-3) and ends at \(x=-2\) (open). Then the next horizontal segment: starts at \(x=-2\) (filled? Wait, no, the dot at \(x=-1\) is filled. Wait, the x - value of - 1: the filled circle is at \(x=-1\), what's its y - value? The y - axis: the line at \(y=-1\) is between \(y=-2\) and \(y = 0\). Wait, no, maybe I made a mistake. Wait, the options are - 3, - 1, 0, 1. Wait, let's look again.

Wait, the graph: the segment that includes \(x=-1\): the filled dot at \(x=-1\) has a y - value of - 1? Wait, no, maybe the segment from \(x=-2\) (filled) to \(x = 1\) (open) with y=-1? Wait, no, the dot at \(x=-1\) is filled, and its y - coordinate is - 1? Wait, but let's check the options. Wait, maybe I misread. Wait, the correct way: for a step function, the value at a point is the y - value of the filled circle (the closed dot) at that x. So at \(x=-1\), we look for the closed dot at \(x=-1\). From the graph, the closed dot at \(x=-1\) is at \(y=-1\)? Wait, no, wait, the y - axis: the first segment (left) is at y=-3, then the next at y=-1? Wait, no, the dot at \(x=-1\) is at y=-1? Wait, but the options include - 1. Wait, but let's check again.

Wait, maybe I made a mistake. Let's re - analyze the graph:

- The left - most horizontal line: y = - 3, from x=-4 (closed) to x=-2 (open).
- Next horizontal line: y=-1, from x=-2 (closed) to x = 1 (open). Wait, at x=-1, which is between - 2 and 1, so the y - value is - 1? But the closed dot at x=-1 is on this line, so f(-1)=-1? But wait, the options: - 3, - 1, 0, 1. Wait, but maybe I misread the graph. Wait, no, wait, the dot at x=-1: looking at the graph, the y - coordinate of the filled dot at x=-1 is - 1? Wait, no, the y - axis: the line at y=-1 is above y=-3. Wait, the first segment (left) is y=-3, then the next is y=-1, then y = 1, then y = 3.

So for x=-1, which is in the interval of the segment that has y=-1 (since x=-1 is between - 2 and 1, assuming the segment is from x=-2 (closed) to x = 1 (open)), so f(-1) is the y - value of that segment, which is - 1? Wait, but let's check the options. The options are - 3, - 1, 0, 1.

Wait, no, wait a second. Wait, the dot at x=-1: is it at y=-1? Wait, the user's graph: the vertical line at x=-1, the filled dot is at y=-1? Or is it at y=-3? No, the left - most segment is y=-3 (from x=-4 to x=-2), then the next segment (from x=-2 to x = 1) is y=-1. So x=-1 is in [-2,1), so f(-1) is - 1? But wait, the options have - 1 as an option. But wait, maybe I made a mistake. Wait, let's look at the graph again. The dot at x=-1: the filled circle. Let's see the coordinates: x=-1, y=-1? Or x=-1, y=-3? No, the y - axis: the first segment (left) is at y=-3 (lower), then the next is at y=-1 (higher), then y = 1, then y = 3. So the segment with y=-1 is between x=-2 and x = 1 (open at x = 1). So x=-1 is in that interval, so f(-1)=-1.

Step1: Identify the x - value

We need to find \(f(-1)\), so we locate \(x=-1\) on the x - axis of the graph of the step function \(f(x)\).

Step2: Determine the function value at \(x=-1\)

For a step function, the value of the function at a point \(x = a\) is the y - coordinate of the filled (closed) circle at \(x=a\) (or the y - coordinate of the horizontal segment that contains \(x=a\)). Looking at the graph, at \(x=-1\), the filled circle (closed dot) has a y - coordinate of \(-1\). So \(f(-1)=-1\).

Answer:

\(\boldsymbol{-1}\) (corresponding to the option " - 1")