To find the coordinate point with an open circle in the given piecewise function, we need to identify the points at which the function changes from one piece to another.
The piecewise function is defined as follows:
- For \( x > 4 \): \( f(x) = 2x + 2 \)
- For \( x \leq 2 \): \( f(x) = \frac{1}{2}x + 1 \)
This means:
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At \( x = 2 \), the function switches from \( f(x) = \frac{1}{2}x + 1 \) to the other piece, but \( x = 2 \) is included in the second piece (since it says \( x \leq 2 \)), hence there will not be an open circle here.
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At \( x = 4 \), the function switches to the first piece \( f(x) = 2x + 2 \), but \( x = 4 \) is not included in the first piece (it is defined only for \( x > 4 \)). Therefore, this means that at \( x = 4 \), there will be an open circle.
To find the coordinate at \( x = 4 \):
- In the first piece for \( x > 4 \), we have \( f(4) \) is undefined, which indicates an open circle.
Now, let's compute the function's value as \( x \) approaches 4 from the right (using the first piece): \[ f(4) = 2(4) + 2 = 8 + 2 = 10 \]
So, the point that will have an open circle is at \( (4, 10) \).
Thus, the coordinate point that will have an open circle is (4, 10).
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