To determine which coordinate point will have an open circle in the graph of the given piecewise function:
The function is defined as follows:
- \( f(x) = 2x + 2 \) if \( x > 2 \)
- \( f(x) = 2x + 1 \) if \( x \leq 2 \)
The critical point is \( x = 2 \) because it is where the definition of the function changes.
- For \( x = 2 \), we look at both parts of the piecewise function:
- For \( x \leq 2 \): \( f(2) = 2(2) + 1 = 4 + 1 = 5 \).
- For \( x > 2 \): The function is not defined for exactly \( x = 2 \), so we do not get a value from that part.
Since the function value at \( x = 2 \) is defined, it will be a solid point on the graph at \( (2, 5) \). However, since for \( x > 2 \) the function continues and \( 2x + 2 \) will produce values for \( x \) just above \( 2 \), there will be an open circle at \( (2, 5) \) to indicate that the function does not include that point.
Thus, the coordinate point that will have an open circle is:
(2, 5)