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Three lines, a solid line, a dotted line, and a dashed line, are drawn on a coordinate plane. The x-axis ranges from negative 4 to 4 in 1-unit increments, and the y-axis ranges from negative 3 to 5 in 1-unit increments.

Which option represents the piecewise function of the graph?

(1 point)
Responses

A. f(x)=⎧⎩⎨⎪⎪−2,3x+1,1, −3<x<−1−1<x≤11<x<3
f left parenthesis x right parenthesis equals left brace Start 3 by 2 Matrix 1st Row 1st Column negative 2 comma 2nd Column negative 3 less than x less than negative 1 2nd Row 1st Column 3 x plus 1 comma 2nd Column negative 1 less than x less-than-or-equal-to 1 3rd Row 1st Column 1 comma 2nd Column 1 less than x less than 3 End Matrix

B. f(x)=⎧⎩⎨⎪⎪−2,3x+1,1, −3≤x<−1−1≤x<11≤x≤3
f left parenthesis x right parenthesis equals left brace Start 3 by 2 Matrix 1st Row 1st Column negative 2 comma 2nd Column negative 3 less-than-or-equal-to x less than negative 1 2nd Row 1st Column 3 x plus 1 comma 2nd Column negative 1 less-than-or-equal-to x less than 1 3rd Row 1st Column 1 comma 2nd Column 1 less-than-or-equal-to x less-than-or-equal-to 3 End Matrix

C. f(x)=⎧⎩⎨⎪⎪−2,3x+1,x, −3≤x<−1−1≤x<11≤x≤3
f left parenthesis x right parenthesis equals left brace Start 3 by 2 Matrix 1st Row 1st Column negative 2 comma 2nd Column negative 3 less-than-or-equal-to x less than negative 1 2nd Row 1st Column 3 x plus 1 comma 2nd Column negative 1 less-than-or-equal-to x less than 1 3rd Row 1st Column x comma 2nd Column 1 less-than-or-equal-to x less-than-or-equal-to 3 End Matrix

D. f(x)=⎧⎩⎨⎪⎪−2.3x+1,1, x<−1−1≤x<11≤x
f left parenthesis x right parenthesis equals left brace Start 3 by 2 Matrix 1st Row 1st Column negative 2 comma 2nd Column x less than negative 1 2nd Row 1st Column 3 x plus 1 comma 2nd Column negative 1 less-than-or-equal-to x less than 1 3rd Row 1st Column 1 comma 2nd Column 1 less-than-or-equal-to x End Matrix

1 answer

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:

  1. Solid Line: This typically indicates a defined value, which might be \(f(x) = -2\) for the interval where \(-3 \leq x < -1\).
  2. Dotted Line: This might indicate a linear function (such as \(f(x) = 3x + 1\)) which often covers the interval from \(-1 \leq x < 1\).
  3. 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.