To sketch the graph of the piecewise function [tex]\( g(x) \)[/tex], let's analyze each part of the piecewise definition separately, and then combine them on the same graph.
The function [tex]\( g(x) \)[/tex] is defined as follows:
[tex]\[ g(x) = \begin{cases}
3x + 4 & , \quad -4 \leq x < 0 \\
-4 & , \quad 0 \leq x \leq 6
\end{cases} \][/tex]
### 1. Analyzing [tex]\( 3x + 4 \)[/tex] for [tex]\( -4 \leq x < 0 \)[/tex]
- Slope and Intercept: The function [tex]\( 3x + 4 \)[/tex] is a linear function with a slope of 3 and a y-intercept of 4.
- Endpoints:
- When [tex]\( x = -4 \)[/tex]:
[tex]\[ g(-4) = 3(-4) + 4 = -12 + 4 = -8 \][/tex]
So, the point [tex]\((-4, -8)\)[/tex] is on the graph, and because [tex]\( x = -4 \)[/tex] is included in the interval, this point will be filled in.
- When [tex]\( x \)[/tex] approaches 0 from the left ([tex]\( x \to 0^- \)[/tex]):
[tex]\[ g(0^-) = 3(0) + 4 = 4 \][/tex]
However, since [tex]\( x = 0 \)[/tex] is not included in this part of the piecewise function, the point [tex]\((0, 4)\)[/tex] will be an open circle.
### 2. Analyzing [tex]\(-4\)[/tex] for [tex]\( 0 \leq x \leq 6 \)[/tex]
- This part of the function is constant, meaning that [tex]\( g(x) = -4 \)[/tex] for every [tex]\( x \)[/tex] in the interval [tex]\( 0 \leq x \leq 6 \)[/tex].
- Endpoints:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -4 \][/tex]
Since [tex]\( x = 0 \)[/tex] is included in this interval, the point [tex]\((0, -4)\)[/tex] will be a filled-in point.
- When [tex]\( x = 6 \)[/tex]:
[tex]\[ g(6) = -4 \][/tex]
Similarly, since [tex]\( x = 6 \)[/tex] is included, the point [tex]\((6, -4)\)[/tex] will be filled in as well.
### Plotting the Graph
Now, let's put it all together:
1. Line segment for [tex]\( 3x + 4 \)[/tex] on [tex]\( -4 \leq x < 0 \)[/tex]
- Starting point: [tex]\((-4, -8)\)[/tex] (filled circle)
- Ending point: Approaches [tex]\((0, 4)\)[/tex] (open circle)
2. Horizontal line for [tex]\( -4 \)[/tex] on [tex]\( 0 \leq x \leq 6 \)[/tex]
- Starting point: [tex]\((0, -4)\)[/tex] (filled circle)
- Ending point: [tex]\((6, -4)\)[/tex] (filled circle)
### Final Graph Sketch
- For the interval [tex]\( -4 \leq x < 0 \)[/tex], plot the line starting from [tex]\((-4, -8)\)[/tex] to just before [tex]\((0, 4)\)[/tex], marking [tex]\((0, 4)\)[/tex] with an open circle.
- For the interval [tex]\( 0 \leq x \leq 6 \)[/tex], plot a horizontal line starting from [tex]\((0, -4)\)[/tex] (filled circle) to [tex]\((6, -4)\)[/tex] (filled circle).
The combined graph will show a linear segment that transitions into a horizontal line.
