Question

evaluate step functions
google classroom microsoft teams

match each expression with its value.
3 7 -3 undefined
$f(-3.0001)$
$f(3.999)$
$f(4.0001)$
$f(9)$

Explanation:

Step1: Analyze \( f(-3.0001) \)

The \( x \)-value \(-3.0001\) is slightly less than \(-3\). Looking at the graph, the left - most horizontal segment (with solid dots) includes \( x \)-values less than \(-3\) (since the right end of that segment is at \( x = - 3\) with a solid dot? Wait, no, looking at the graph, the segment with solid dots on the left: let's see the \( x \)-axis. The left segment has solid dots, and when \( x\) is around \(-8\) to \(-3\) (the right end is at \( x=-3\) with a solid dot? Wait, no, the graph: the left horizontal line (with solid dots) is from, say, \( x=-8\) (or more left) to \( x = - 3\) (solid dot). Then, for \( x=-3.0001\), which is less than \(-3\), we look at the segment that includes \( x < - 3\). Wait, no, maybe I misread. Wait, the left segment: the solid dots, so the domain for that segment is \( x\leq - 3\)? Wait, no, the left segment has a solid dot at the left (maybe \( x=-8\)) and a solid dot at \( x = - 3\). Then, the middle segment (with open circle at \( x=-2\) and solid dot at \( x = 5\)? Wait, no, let's re - examine.

Wait, the graph:

- The bottom - right segment: open circles at \( x = 4\) and \( x = 8\), so domain \( 4<x<8\), \( y=-3\).

- The middle - top segment: open circle at \( x=-2\) and solid dot at \( x = 5\), \( y = 7\) (since it's at \( y = 7\)).

- The left segment: solid dots, from, say, \( x=-8\) to \( x=-3\), \( y = 3\) (since it's at \( y = 3\)).

So for \( f(-3.0001)\): \( x=-3.0001\) is less than \(-3\). The left segment (with solid dots) includes \( x\leq - 3\) (since the right end is at \( x=-3\) with a solid dot? Wait, no, if the right end of the left segment is at \( x=-3\) (solid dot), then \( x=-3.0001\) is in the domain of the left segment. The \( y\) - value for the left segment is \( 3\)? Wait, no, wait the left segment is at \( y = 3\)? Wait, the left segment is at \( y = 3\) (since the \( y\) - coordinate of that segment is 3). Wait, no, looking at the \( y\) - axis: the left segment is at \( y = 3\) (the horizontal line at \( y = 3\) with solid dots), the middle - top segment is at \( y = 7\) (horizontal line with open circle at \( x=-2\) and solid dot at \( x = 5\)), and the bottom - right segment is at \( y=-3\) (open circles at \( x = 4\) and \( x = 8\)).

Wait, maybe I made a mistake. Let's correct:

- Left segment: solid dots, \( y = 3\), domain \( x\leq - 3\) (since the right end is a solid dot at \( x=-3\)).

- Middle segment: open circle at \( x=-2\), solid dot at \( x = 5\), \( y = 7\), domain \(-2<x\leq5\).

- Bottom - right segment: open circles at \( x = 4\) and \( x = 8\), \( y=-3\), domain \( 4<x<8\).

So \( f(-3.0001)\): \( x=-3.0001\leq - 3\), so it's in the left segment. The \( y\) - value of the left segment is \( 3\)? Wait, no, the left segment is at \( y = 3\)? Wait, the left segment is at \( y = 3\) (the horizontal line at \( y = 3\) with solid dots). So \( f(-3.0001)=3\)? Wait, no, wait the left segment: the \( y\) - coordinate is 3? Wait, the left segment is at \( y = 3\) (the horizontal line with solid dots, \( y = 3\)).

Wait, no, maybe the left segment is at \( y = 3\), middle at \( y = 7\), bottom - right at \( y=-3\).

So \( f(-3.0001)\): \( x=-3.0001\) is in the left segment (since \( x\leq - 3\)), so \( f(-3.0001)=3\)? Wait, no, wait the left segment's \( y\) - value: looking at the graph, the left segment is at \( y = 3\) (the horizontal line at \( y = 3\) with solid dots).

Step2: Analyze \( f(3.999) \)

\( x = 3.999\). Let's see the domains:

- Left segment: \( x\leq - 3\), not applicable.

- Middle segment: \(-2<x\leq5\), \( y = 7\). Since \( 3.999\) is between \(-2\) and \( 5\) (because \(-2<3.999\leq5\)), so \( f(3.999)=7\).

Step3: Analyze \( f(4.0001) \)

\( x = 4.0001\). The bottom - right segment has domain \( 4<x<8\), \( y=-3\). Since \( 4.0001>4\) and \( 4.0001<8\), so \( f(4.0001)=-3\).

Step4: Analyze \( f(9) \)

\( x = 9\). Let's check the domains:

- Left segment: \( x\leq - 3\), no.

- Middle segment: \(-2<x\leq5\), no (since \( 9>5\)).

- Bottom - right segment: \( 4<x<8\), no (since \( 9>8\)). So there is no segment that includes \( x = 9\) in its domain. So \( f(9)\) is undefined.

Answer:

• \( f(-3.0001)=3\)

• \( f(3.999)=7\)

• \( f(4.0001)=-3\)

• \( f(9)=\text{Undefined}\)