Question
The graph shows a line from (1,1) to (-1,-3) And (-1,1) to (-3,1)
Create a piecewise function that represents the graph.
(1 point)
Responses
f(x)={1,−3≤x<−1
2x,−1≤x≤1
f left parenthesis x right parenthesis equals left brace Start 2 by 2 Matrix 1st Row 1st Column 1 comma 2nd Column negative 3 less-than-or-equal-to x less than negative 1 2nd Row 1st Column 2 x comma 2nd Column negative 1 less-than-or-equal-to x less-than-or-equal-to 1 End Matrix
f(x)={1,−3<x≤−1
2x−1,−1<x<1
f left parenthesis x right parenthesis equals left brace Start 2 by 2 Matrix 1st Row 1st Column 1 comma 2nd Column negative 3 less than x less-than-or-equal-to negative 1 2nd Row 1st Column 2 x minus 1 comma 2nd Column negative 1 less than x less than 1 End Matrix
f(x)={1,−3≤x<−1
2x−1,−1≤x≤1
f left parenthesis x right parenthesis equals left brace Start 2 by 2 Matrix 1st Row 1st Column 1 comma 2nd Column negative 3 less-than-or-equal-to x less than negative 1 2nd Row 1st Column 2 x minus 1 comma 2nd Column negative 1 less-than-or-equal-to x less-than-or-equal-to 1 End Matrix
f(x)={x+1,−3≤x<−1
2x−1,−1≤x≤1
Create a piecewise function that represents the graph.
(1 point)
Responses
f(x)={1,−3≤x<−1
2x,−1≤x≤1
f left parenthesis x right parenthesis equals left brace Start 2 by 2 Matrix 1st Row 1st Column 1 comma 2nd Column negative 3 less-than-or-equal-to x less than negative 1 2nd Row 1st Column 2 x comma 2nd Column negative 1 less-than-or-equal-to x less-than-or-equal-to 1 End Matrix
f(x)={1,−3<x≤−1
2x−1,−1<x<1
f left parenthesis x right parenthesis equals left brace Start 2 by 2 Matrix 1st Row 1st Column 1 comma 2nd Column negative 3 less than x less-than-or-equal-to negative 1 2nd Row 1st Column 2 x minus 1 comma 2nd Column negative 1 less than x less than 1 End Matrix
f(x)={1,−3≤x<−1
2x−1,−1≤x≤1
f left parenthesis x right parenthesis equals left brace Start 2 by 2 Matrix 1st Row 1st Column 1 comma 2nd Column negative 3 less-than-or-equal-to x less than negative 1 2nd Row 1st Column 2 x minus 1 comma 2nd Column negative 1 less-than-or-equal-to x less-than-or-equal-to 1 End Matrix
f(x)={x+1,−3≤x<−1
2x−1,−1≤x≤1
Answers
GPT-4o mini
To create a piecewise function that represents the given graph, we need to analyze the segments and their corresponding equations more closely.
- The line from \((-1, -3)\) to \((1, 1)\) can be represented by a linear function. The slope is calculated as:
\[
\text{slope} = \frac{1 - (-3)}{1 - (-1)} = \frac{4}{2} = 2
\]
Using the point-slope form \(y - y_1 = m(x - x_1)\) with the point \((-1, -3)\):
\[
y - (-3) = 2(x - (-1)) \\
y + 3 = 2(x + 1) \\
y = 2x + 2 - 3 \\
y = 2x - 1
\]
- The horizontal segment from \((-3, 1)\) to \((-1, 1)\) is represented by the constant function \(f(x) = 1\).
Now, we can define the piecewise function based on these observations:
- For \(-3 \leq x < -1\), the function is \(f(x) = 1\).
- For \(-1 \leq x \leq 1\), the function is \(f(x) = 2x - 1\).
Putting this all together, the piecewise function is:
\[
f(x) =
\begin{cases}
1 & \text{for } -3 \leq x < -1 \\
2x - 1 & \text{for } -1 \leq x \leq 1
\end{cases}
\]
Based on the choices you provided, the correct one that closely matches our derived piecewise function is:
**f(x)={1,−3≤x<−1, 2x-1,−1≤x≤1}**.
- The line from \((-1, -3)\) to \((1, 1)\) can be represented by a linear function. The slope is calculated as:
\[
\text{slope} = \frac{1 - (-3)}{1 - (-1)} = \frac{4}{2} = 2
\]
Using the point-slope form \(y - y_1 = m(x - x_1)\) with the point \((-1, -3)\):
\[
y - (-3) = 2(x - (-1)) \\
y + 3 = 2(x + 1) \\
y = 2x + 2 - 3 \\
y = 2x - 1
\]
- The horizontal segment from \((-3, 1)\) to \((-1, 1)\) is represented by the constant function \(f(x) = 1\).
Now, we can define the piecewise function based on these observations:
- For \(-3 \leq x < -1\), the function is \(f(x) = 1\).
- For \(-1 \leq x \leq 1\), the function is \(f(x) = 2x - 1\).
Putting this all together, the piecewise function is:
\[
f(x) =
\begin{cases}
1 & \text{for } -3 \leq x < -1 \\
2x - 1 & \text{for } -1 \leq x \leq 1
\end{cases}
\]
Based on the choices you provided, the correct one that closely matches our derived piecewise function is:
**f(x)={1,−3≤x<−1, 2x-1,−1≤x≤1}**.