Question
Which of the following piecewise functions matches the graph?
(1 point)
Responses

f(x)={2−x if x≤0x−5 if x>0
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row 2 minus x if x less-than-or-equal-to 0 2nd Row x minus 5 if x greater than 0 End Rows

f(x)={x+5 if x≤−22x+3 if x>−2
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row x plus 5 if x less-than-or-equal-to negative 2 2nd Row 2 x plus 3 if x greater than negative 2 End Rows

f(x)={−x−1 if x<1−x+2 if x≥1
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row negative x minus 1 if x less than 1 2nd Row negative x plus 2 if x greater-than-or-equal-to 1 End Rows

f(x)={−5 if x<0−x+1 if x≥0
(1 point)
Responses

f(x)={2−x if x≤0x−5 if x>0
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row 2 minus x if x less-than-or-equal-to 0 2nd Row x minus 5 if x greater than 0 End Rows

f(x)={x+5 if x≤−22x+3 if x>−2
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row x plus 5 if x less-than-or-equal-to negative 2 2nd Row 2 x plus 3 if x greater than negative 2 End Rows

f(x)={−x−1 if x<1−x+2 if x≥1
f left parenthesis x right parenthesis equals left brace Start Rows 1st Row negative x minus 1 if x less than 1 2nd Row negative x plus 2 if x greater-than-or-equal-to 1 End Rows

f(x)={−5 if x<0−x+1 if x≥0
Answers
GPT-4o mini
To determine which piecewise function matches the graph, we need to analyze the behavior of each function based on its defined conditions and values.
Let's break down each function:
1. **Function 1**:
\[
f(x) =
\begin{cases}
2 - x & \text{if } x \leq 0 \\
x - 5 & \text{if } x > 0
\end{cases}
\]
- For \(x \leq 0\), this is a decreasing line starting from \(f(0) = 2\) and going downwards as x decreases.
- For \(x > 0\), this is an increasing line starting from \(f(0) = -5\) at \(x=0\) and moving upwards.
2. **Function 2**:
\[
f(x) =
\begin{cases}
x + 5 & \text{if } x \leq -2 \\
2x + 3 & \text{if } x > -2
\end{cases}
\]
- For \(x \leq -2\), this is a line that shifts up by 5.
- For \(x > -2\), it starts at a certain point \(f(-2) = -1\) and increases.
3. **Function 3**:
\[
f(x) =
\begin{cases}
-x - 1 & \text{if } x < 1 \\
-x + 2 & \text{if } x \geq 1
\end{cases}
\]
- For \(x < 1\), this is a decreasing line that hits \(-2\) at \(1\).
- For \(x \geq 1\), it continues from the point at \(1\) upwards.
4. **Function 4**:
\[
f(x) =
\begin{cases}
-5 & \text{if } x < 0 \\
-x + 1 & \text{if } x \geq 0
\end{cases}
\]
- For \(x < 0\), the function is constant at \(-5\).
- For \(x \geq 0\), the line has a y-intercept of \(1\) at \(x=0\) and has a downward slope.
To determine which function matches a specific graph, you should check for key features such as intercepts, slopes, values at threshold points, and intervals where the functions behave differently.
If you have a description or a visual representation of the graph, you can use this reasoning to match the appropriate function.
Let's break down each function:
1. **Function 1**:
\[
f(x) =
\begin{cases}
2 - x & \text{if } x \leq 0 \\
x - 5 & \text{if } x > 0
\end{cases}
\]
- For \(x \leq 0\), this is a decreasing line starting from \(f(0) = 2\) and going downwards as x decreases.
- For \(x > 0\), this is an increasing line starting from \(f(0) = -5\) at \(x=0\) and moving upwards.
2. **Function 2**:
\[
f(x) =
\begin{cases}
x + 5 & \text{if } x \leq -2 \\
2x + 3 & \text{if } x > -2
\end{cases}
\]
- For \(x \leq -2\), this is a line that shifts up by 5.
- For \(x > -2\), it starts at a certain point \(f(-2) = -1\) and increases.
3. **Function 3**:
\[
f(x) =
\begin{cases}
-x - 1 & \text{if } x < 1 \\
-x + 2 & \text{if } x \geq 1
\end{cases}
\]
- For \(x < 1\), this is a decreasing line that hits \(-2\) at \(1\).
- For \(x \geq 1\), it continues from the point at \(1\) upwards.
4. **Function 4**:
\[
f(x) =
\begin{cases}
-5 & \text{if } x < 0 \\
-x + 1 & \text{if } x \geq 0
\end{cases}
\]
- For \(x < 0\), the function is constant at \(-5\).
- For \(x \geq 0\), the line has a y-intercept of \(1\) at \(x=0\) and has a downward slope.
To determine which function matches a specific graph, you should check for key features such as intercepts, slopes, values at threshold points, and intervals where the functions behave differently.
If you have a description or a visual representation of the graph, you can use this reasoning to match the appropriate function.