Question

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.
the solid line goes from (1,4) to (-1,-2) the dotted line goes from (-3,-2) to (-1,-2) the dashed line goes from (1,1) to (3,1)
Which option represents the piecewise function of the graph?

(1 point)
Responses

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

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

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

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

Answer 1

To write the piecewise function based on the given line segments, we can analyze each line:

1. **Solid line going from (1,4) to (-1,-2)**:
This line segment has endpoints (1, 4) and (-1, -2). We first calculate the slope \( m \) using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 4}{-1 - 1} = \frac{-6}{-2} = 3
\]
The equation can be found using the point-slope form \( y - y_1 = m(x - x_1) \):
\[
y - 4 = 3(x - 1) \implies y = 3x + 1
\]
Hence, this section applies for \(-1 < x \leq 1\).

2. **Dotted line going from (-3,-2) to (-1,-2)**:
This is a horizontal line at \( y = -2 \), applying for \(-3 \leq x < -1\).

3. **Dashed line going from (1,1) to (3,1)**:
This is also a horizontal line at \( y = 1 \) applicable for \( 1 < x \leq 3\).

Now, we can put these together in a piecewise function format:

- For \(-3 \leq x < -1\), \(f(x) = -2\)
- For \(-1 < x \leq 1\), \(f(x) = 3x + 1\)
- For \(1 < x \leq 3\), \(f(x) = 1\)

Putting it all together, the piecewise function can be expressed as:
\[
f(x) =
\begin{cases}
-2 & \text{if } -3 \leq x < -1 \\
3x + 1 & \text{if } -1 < x \leq 1 \\
1 & \text{if } 1 < x \leq 3
\end{cases}
\]

Based on the given options, the correct answer is:

**Option 2**:
\[
f(x)=\begin{cases} -2 & \text{if } -3 \leq x < -1 \\ 3x + 1 & \text{if } -1 < x \leq 1 \\ 1 & \text{if } 1 < x \leq 3 \end{cases}
\]