To find the interval(s) over which the function 8-(x-3)^2 is increasing, we need to find the critical points of the function and test the sign of the derivative on either side of these points.
The derivative of the function 8-(x-3)^2 is -2(x-3).
Setting the derivative equal to zero to find the critical points:
-2(x-3) = 0
x = 3
Testing the sign of the derivative on either side of x = 3:
-2(x-3) is negative for x < 3, meaning the function is decreasing on the interval (-infinity, 3).
-2(x-3) is positive for x > 3, meaning the function is increasing on the interval (3, infinity).
Therefore, the interval over which the function 8-(x-3)^2 is increasing is (3, infinity).