To graph the piecewise function defined as:
\[ f(x) = \begin{cases} 2x - 1 & \text{for } x > -1 \ x + 5 & \text{for } x \leq -1 \end{cases} \]
we will break it down.
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For \( x > -1 \):
- The function is \( f(x) = 2x - 1 \).
- This is a straight line with a slope of 2 and a y-intercept of -1.
- We will need to plot points for \( x > -1 \) and note that this line does not include the point at \( x = -1 \).
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For \( x \leq -1 \):
- The function is \( f(x) = x + 5 \).
- This is also a straight line with a slope of 1 and a y-intercept of 5.
- We will include the point at \( x = -1 \) in this portion of the graph.
Next, let's calculate some key points for both parts:
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For \( f(x) = 2x - 1 \):
- At \( x = 0 \): \( f(0) = 2(0) - 1 = -1 \)
- At \( x = -0.5 \): \( f(-0.5) = 2(-0.5) - 1 = -2 \)
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For \( f(x) = x + 5 \):
- At \( x = -1 \): \( f(-1) = -1 + 5 = 4 \)
- At \( x = -2 \): \( f(-2) = -2 + 5 = 3 \)
Now we can summarize the information:
- For \( x > -1 \): A line through points like \( (-0.5, -2) \) and \( (0, -1) \).
- For \( x \leq -1 \): A line that includes \( (-1, 4) \) and point \( (-2, 3) \).
Given this information, we can identify the characteristics of the graph segments:
- The line \( f(x) = 2x - 1 \) will have an open circle at \( (-1, 1) \) and continue to rise steeply to the right.
- The line \( f(x) = x + 5 \) will be drawn solidly from \( (-1, 4) \).
Choosing the Correct Graph
To select the correct graph among the options, look for:
- An open circle at the point corresponding to \( x = -1 \) on the line \( 2x - 1 \).
- A solid point on the line \( x + 5 \) at \( (-1, 4) \).
Take a look at the options A, B, C, and D. The correct graph will show these characteristics.
If you are able to analyze the graphs, choose the one that matches this description!
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