To determine which option correctly represents the piecewise function for the graph described, we need to analyze the function segments and their intervals.
Based on the structure of a piecewise function, each piece corresponds to a specific interval of x-values:
- Solid Line: This typically indicates a defined value, which might be \(f(x) = -2\) for the interval where \(-3 \leq x < -1\).
- Dotted Line: This might indicate a linear function (such as \(f(x) = 3x + 1\)) which often covers the interval from \(-1 \leq x < 1\).
- Dashed Line: This line may represent a constant function, perhaps \(f(x) = 1\) for the interval \(1 \leq x \leq 3\).
Now, let's check each option:
A.
- \(f(x) = -2\) for \(-3 < x < -1\) (does not include endpoints)
- \(f(x) = 3x + 1\) for \(-1 < x \leq 1\)
- \(f(x) = 1\) for \(1 < x < 3\) (does not include endpoint)
B.
- \(f(x) = -2\) for \(-3 \leq x < -1\) (includes -3)
- \(f(x) = 3x + 1\) for \(-1 \leq x < 1\) (includes -1)
- \(f(x) = 1\) for \(1 \leq x \leq 3\) (includes both 1 and 3)
C.
- \(f(x) = -2\) for \(-3 \leq x < -1\) (includes -3)
- \(f(x) = 3x + 1\) for \(-1 \leq x < 1\) (includes -1)
- \(f(x) = x\) for \(1 \leq x \leq 3\) (includes both 1 and 3)
D.
- \(f(x) = -2\) for \(x < -1\) (does not restrict to -3)
- \(f(x) = 3x + 1\) for \(-1 \leq x < 1\)
- \(f(x) = 1\) for \(1 \leq x\) (no upper limit).
Conclusion:
Considering the definitions and the continuity at the boundaries (whether they include or exclude endpoints), option B seems like the most appropriate representation because it properly defines the intervals, including or excluding boundaries as needed:
B. \[ f(x) = \begin{cases} -2 & , -3 \leq x < -1 \ 3x + 1 & , -1 \leq x < 1 \ 1 & , 1 \leq x \leq 3 \end{cases} \]
Thus, the answer is B.