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Sagot :
Sure, let's go step by step to graph the piecewise function [tex]\( f(x) \)[/tex] and determine the corresponding intervals:
[tex]\[ f(x) = \begin{cases} 6 & \text{if } -5 < x \leq -1 \\ 4 & \text{if } -1 < x \leq 2 \\ 2 & \text{if } 2 < x \leq 5 \end{cases} \][/tex]
1. Graphing the first interval: [tex]\( -5 < x \leq -1 \)[/tex]
- For values [tex]\( x \)[/tex] from [tex]\( -5 \)[/tex] (not inclusive) to [tex]\( -1 \)[/tex] (inclusive), [tex]\( f(x) = 6 \)[/tex].
- Plot a horizontal line at [tex]\( y = 6 \)[/tex] from [tex]\( x = -5 \)[/tex] to [tex]\( x = -1 \)[/tex].
- Place an open circle at [tex]\( ( -5, 6 ) \)[/tex] to indicate that [tex]\( x = -5 \)[/tex] is not included.
- Place a closed circle at [tex]\( ( -1, 6 ) \)[/tex] to indicate that [tex]\( x = -1 \)[/tex] is included.
2. Graphing the second interval: [tex]\( -1 < x \leq 2 \)[/tex]
- For values [tex]\( x \)[/tex] from [tex]\( -1 \)[/tex] (not inclusive) to [tex]\( 2 \)[/tex] (inclusive), [tex]\( f(x) = 4 \)[/tex].
- Plot a horizontal line at [tex]\( y = 4 \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex].
- Place an open circle at [tex]\( ( -1, 4 ) \)[/tex] to indicate that [tex]\( x = -1 \)[/tex] is not included.
- Place a closed circle at [tex]\( ( 2, 4 ) \)[/tex] to indicate that [tex]\( x = 2 \)[/tex] is included.
3. Graphing the third interval: [tex]\( 2 < x \leq 5 \)[/tex]
- For values [tex]\( x \)[/tex] from [tex]\( 2 \)[/tex] (not inclusive) to [tex]\( 5 \)[/tex] (inclusive), [tex]\( f(x) = 2 \)[/tex].
- Plot a horizontal line at [tex]\( y = 2 \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 5 \)[/tex].
- Place an open circle at [tex]\( ( 2, 2 ) \)[/tex] to indicate that [tex]\( x = 2 \)[/tex] is not included.
- Place a closed circle at [tex]\( ( 5, 2 ) \)[/tex] to indicate that [tex]\( x = 5 \)[/tex] is included.
By following these steps, you should get the piecewise graph:
- A horizontal segment from [tex]\( x = -5 \)[/tex] to [tex]\( x = -1 \)[/tex] at [tex]\( y = 6 \)[/tex] with an open circle on the far left and a closed circle on the far right.
- A horizontal segment from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex] at [tex]\( y = 4 \)[/tex] with an open circle on the far left and a closed circle on the far right.
- A horizontal segment from [tex]\( x = 2 \)[/tex] to [tex]\( x = 5 \)[/tex] at [tex]\( y = 2 \)[/tex] with an open circle on the far left and a closed circle on the far right.
Now, let's discuss the intervals and the corresponding values to confirm the graph is correct.
So the graph matches the description as given above for each piecewise interval of the function [tex]\( f(x) \)[/tex].
[tex]\[ f(x) = \begin{cases} 6 & \text{if } -5 < x \leq -1 \\ 4 & \text{if } -1 < x \leq 2 \\ 2 & \text{if } 2 < x \leq 5 \end{cases} \][/tex]
1. Graphing the first interval: [tex]\( -5 < x \leq -1 \)[/tex]
- For values [tex]\( x \)[/tex] from [tex]\( -5 \)[/tex] (not inclusive) to [tex]\( -1 \)[/tex] (inclusive), [tex]\( f(x) = 6 \)[/tex].
- Plot a horizontal line at [tex]\( y = 6 \)[/tex] from [tex]\( x = -5 \)[/tex] to [tex]\( x = -1 \)[/tex].
- Place an open circle at [tex]\( ( -5, 6 ) \)[/tex] to indicate that [tex]\( x = -5 \)[/tex] is not included.
- Place a closed circle at [tex]\( ( -1, 6 ) \)[/tex] to indicate that [tex]\( x = -1 \)[/tex] is included.
2. Graphing the second interval: [tex]\( -1 < x \leq 2 \)[/tex]
- For values [tex]\( x \)[/tex] from [tex]\( -1 \)[/tex] (not inclusive) to [tex]\( 2 \)[/tex] (inclusive), [tex]\( f(x) = 4 \)[/tex].
- Plot a horizontal line at [tex]\( y = 4 \)[/tex] from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex].
- Place an open circle at [tex]\( ( -1, 4 ) \)[/tex] to indicate that [tex]\( x = -1 \)[/tex] is not included.
- Place a closed circle at [tex]\( ( 2, 4 ) \)[/tex] to indicate that [tex]\( x = 2 \)[/tex] is included.
3. Graphing the third interval: [tex]\( 2 < x \leq 5 \)[/tex]
- For values [tex]\( x \)[/tex] from [tex]\( 2 \)[/tex] (not inclusive) to [tex]\( 5 \)[/tex] (inclusive), [tex]\( f(x) = 2 \)[/tex].
- Plot a horizontal line at [tex]\( y = 2 \)[/tex] from [tex]\( x = 2 \)[/tex] to [tex]\( x = 5 \)[/tex].
- Place an open circle at [tex]\( ( 2, 2 ) \)[/tex] to indicate that [tex]\( x = 2 \)[/tex] is not included.
- Place a closed circle at [tex]\( ( 5, 2 ) \)[/tex] to indicate that [tex]\( x = 5 \)[/tex] is included.
By following these steps, you should get the piecewise graph:
- A horizontal segment from [tex]\( x = -5 \)[/tex] to [tex]\( x = -1 \)[/tex] at [tex]\( y = 6 \)[/tex] with an open circle on the far left and a closed circle on the far right.
- A horizontal segment from [tex]\( x = -1 \)[/tex] to [tex]\( x = 2 \)[/tex] at [tex]\( y = 4 \)[/tex] with an open circle on the far left and a closed circle on the far right.
- A horizontal segment from [tex]\( x = 2 \)[/tex] to [tex]\( x = 5 \)[/tex] at [tex]\( y = 2 \)[/tex] with an open circle on the far left and a closed circle on the far right.
Now, let's discuss the intervals and the corresponding values to confirm the graph is correct.
So the graph matches the description as given above for each piecewise interval of the function [tex]\( f(x) \)[/tex].
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