To understand the graph of the piecewise function [tex]\( g(x) \)[/tex], let's analyze each piece of the function and see how it contributes to the overall graph:
1. For [tex]\(-4 \leq x < 0\)[/tex]:
[tex]\[
g(x) = 3x + 4
\][/tex]
This is a linear equation with a slope of 3 and a y-intercept at 4. The graph of this part is a line that starts from [tex]\( x = -4 \)[/tex] and goes up to [tex]\( x = 0 \)[/tex] (but not inclusive of 0).
- When [tex]\( x = -4 \)[/tex]:
[tex]\[
g(-4) = 3(-4) + 4 = -12 + 4 = -8
\][/tex]
So, the point (-4, -8) is on the graph.
- When [tex]\( x \)[/tex] approaches 0 (from the left),
[tex]\[
g(0^-) = 3(0) + 4 = 4
\][/tex]
At [tex]\( x = 0 \)[/tex], the value is approaching 4, but since [tex]\( x = 0 \)[/tex] is exclusive in this segment, the graph does not actually touch (0, 4) in this part.
2. For [tex]\( 0 \leq x \leq 6 \)[/tex]:
[tex]\[
g(x) = -4
\][/tex]
This part of the function is a constant value of -4, meaning the graph is a horizontal line at [tex]\( y = -4 \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 6 \)[/tex].
Combining these:
- For [tex]\( -4 \leq x < 0 \)[/tex], the graph is a straight line that starts at the point (-4, -8) and goes up to, but does not include, the point (0, 4).
- For [tex]\( 0 \leq x \leq 6 \)[/tex], the graph is a horizontal line at [tex]\( y = -4 \)[/tex].
### Description of the Graphs:
- If you are looking at graph (A):
- Verify if it has a line that starts from (-4, -8) and rises to just below (0, 4).
- Ensure there's a horizontal line at [tex]\( y = -4 \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 6 \)[/tex].
- If you are looking at graph (B):
- Verify if it meets the same criteria: a line from (-4, -8) to just below (0, 4) and then a constant line at [tex]\( y = -4 \)[/tex] from [tex]\( x = 0 \)[/tex] to [tex]\( x = 6 \)[/tex].
Based on the detailed analysis, the graph should meet these specifications. Therefore, the correct graph is the one that matches this description, and the chosen answer is (B).
