To determine which graph corresponds to the piecewise function
[tex]\[
f(x) = \left\{\begin{array}{ll}
\frac{1}{2} x - 2 & \text{if } x \leq 6 \\
-x - 1 & \text{if } x > 6
\end{array}\right.
\][/tex]
we need to analyze each part of the function and find their points and slopes to match them with one of the provided graphs.
### For [tex]\( x \leq 6 \)[/tex]:
The function is [tex]\( f(x) = \frac{1}{2}x - 2 \)[/tex].
1. Calculate the value at the boundary [tex]\( x = 6 \)[/tex]:
[tex]\[
f(6) = \frac{1}{2} \cdot 6 - 2 = 3 - 2 = 1
\][/tex]
So at [tex]\( x = 6 \)[/tex], [tex]\( f(x) \)[/tex] is 1.
2. Calculate the y-intercept:
[tex]\[
\text{Set } x = 0: \quad f(0) = \frac{1}{2} \cdot 0 - 2 = -2
\][/tex]
So at [tex]\( x = 0 \)[/tex], [tex]\( f(x) = -2 \)[/tex].
3. Determine the slope:
[tex]\[
\text{The slope is the coefficient of } x, \text{ which is } \frac{1}{2}.
\][/tex]
### For [tex]\( x > 6 \)[/tex]:
The function is [tex]\( f(x) = -x - 1 \)[/tex].
1. Calculate the value at the boundary [tex]\( x = 6 \)[/tex]:
[tex]\[
f(6) = -6 - 1 = -7
\][/tex]
Since it's for [tex]\( x > 6 \)[/tex], [tex]\( f(6) \)[/tex] is undefined here. However, it gives us a boundary point.
2. Behavior near [tex]\( x = 6 \)[/tex]:
[tex]\[
\text{For values just greater than 6, apply:}
\][/tex]
If [tex]\( x = 7 \)[/tex]:
[tex]\[
f(7) = -7 - 1 = -8
\][/tex]
If [tex]\( x = 8 \)[/tex]:
[tex]\[
f(8) = -8 - 1 = -9
\][/tex]
3. Determine the slope:
[tex]\[
\text{The slope is -1, which means it is decreasing linearly.}
\][/tex]
### Summary:
For the piecewise function:
- For [tex]\( x \leq 6 \)[/tex], it is a line with slope [tex]\( \frac{1}{2} \)[/tex] and passes through points [tex]\( (0, -2) \)[/tex] and [tex]\( (6, 1) \)[/tex].
- For [tex]\( x > 6 \)[/tex], it is a line with slope [tex]\( -1 \)[/tex] and indicates tearing from point [tex]\( (6, -7 )\)[/tex] onward.
With these criterions, try to compare the solution presented in these descriptions accordingly to the graphical representations provided in the choices (A, B, C, D).
Based on the straightforward match with criteria, determine which graph fits this detailing and pick the correct graph. In absence of actual graphical representation, you must visually interpret the discussed slopes and intercepts. Thus, accurately matching them verifies a correct piecewise interpretation.
