Question
Graph the piecewise function.
f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column 2 x minus 1 comma 2nd Column for x greater than minus 1 2nd Row 1st Column x plus 4 comma 2nd Column for x less than or equals minus 1 EndMatrix
Question content area bottom
Part 1
Choose the correct graph below.
A.
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x
y
x y graph
B.
-10
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-10
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x
y
x y graph
C.
-10
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-10
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x
y
x y graph
D.
-10
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x
y
f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column 2 x minus 1 comma 2nd Column for x greater than minus 1 2nd Row 1st Column x plus 4 comma 2nd Column for x less than or equals minus 1 EndMatrix
Question content area bottom
Part 1
Choose the correct graph below.
A.
-10
10
-10
10
x
y
x y graph
B.
-10
10
-10
10
x
y
x y graph
C.
-10
10
-10
10
x
y
x y graph
D.
-10
10
-10
10
x
y
Answers
GPT-4o mini
To graph the piecewise function given:
\[
f(x) = \begin{cases}
2x - 1 & \text{for } x > -1 \\
x + 4 & \text{for } x \leq -1
\end{cases}
\]
we need to analyze both parts of the function separately.
### Step 1: Graph \( f(x) = 2x - 1 \) for \( x > -1 \)
This is a linear function with a slope of 2 and a y-intercept of -1.
1. Find the point where it would intersect the boundary \( x = -1 \):
\[
f(-1) = 2(-1) - 1 = -2 - 1 = -3
\]
However, this point is not included in this part of the function, so we will indicate it with an open circle when we plot it.
2. The line \( y = 2x - 1 \) will continue for values greater than \(-1\).
### Step 2: Graph \( f(x) = x + 4 \) for \( x \leq -1 \)
This is another linear function that has a slope of 1 and a y-intercept of 4.
1. When \( x = -1 \):
\[
f(-1) = -1 + 4 = 3
\]
This point is part of this piece of the function and will be marked with a solid dot.
2. As \( x \) decreases (i.e., moves left), the line continues for values less than or equal to \(-1\).
### Combining these parts
- At \( x = -1 \), the function has two parts:
- From the second segment, it is \( 3 \) (which is included).
- From the first segment, the value approaches \( -3 \) (which is not included).
### Finalizing the Graph
Now, you should have the following characteristics for your graph:
- The line \( y = 2x - 1 \) starting from just right of \(-1\) with an open circle at \((-1, -3)\).
- The line \( y = x + 4 \) starting from the point \((-1, 3)\) and continuing to the left.
With this analysis, you can select the appropriate graph:
- **Look for a graph with:
- An open circle at \((-1, -3)\).
- A closed dot at \((-1, 3)\).
- A positive slope line for \( x \) values greater than \(-1\).
- A positive slope line intersecting at \((-1, 3)\) for \( x \) values less or equal to \(-1\).**
If you have the graphs labeled A, B, C, and D, identify based on these characteristics. If you describe the graphs, I may help more specifically to point out which one fits!
\[
f(x) = \begin{cases}
2x - 1 & \text{for } x > -1 \\
x + 4 & \text{for } x \leq -1
\end{cases}
\]
we need to analyze both parts of the function separately.
### Step 1: Graph \( f(x) = 2x - 1 \) for \( x > -1 \)
This is a linear function with a slope of 2 and a y-intercept of -1.
1. Find the point where it would intersect the boundary \( x = -1 \):
\[
f(-1) = 2(-1) - 1 = -2 - 1 = -3
\]
However, this point is not included in this part of the function, so we will indicate it with an open circle when we plot it.
2. The line \( y = 2x - 1 \) will continue for values greater than \(-1\).
### Step 2: Graph \( f(x) = x + 4 \) for \( x \leq -1 \)
This is another linear function that has a slope of 1 and a y-intercept of 4.
1. When \( x = -1 \):
\[
f(-1) = -1 + 4 = 3
\]
This point is part of this piece of the function and will be marked with a solid dot.
2. As \( x \) decreases (i.e., moves left), the line continues for values less than or equal to \(-1\).
### Combining these parts
- At \( x = -1 \), the function has two parts:
- From the second segment, it is \( 3 \) (which is included).
- From the first segment, the value approaches \( -3 \) (which is not included).
### Finalizing the Graph
Now, you should have the following characteristics for your graph:
- The line \( y = 2x - 1 \) starting from just right of \(-1\) with an open circle at \((-1, -3)\).
- The line \( y = x + 4 \) starting from the point \((-1, 3)\) and continuing to the left.
With this analysis, you can select the appropriate graph:
- **Look for a graph with:
- An open circle at \((-1, -3)\).
- A closed dot at \((-1, 3)\).
- A positive slope line for \( x \) values greater than \(-1\).
- A positive slope line intersecting at \((-1, 3)\) for \( x \) values less or equal to \(-1\).**
If you have the graphs labeled A, B, C, and D, identify based on these characteristics. If you describe the graphs, I may help more specifically to point out which one fits!