where is the graph of ( f(x)=4lfloor x - 3\rfloor+2 ) discontinuous?\nall real numbers\nall integers\nonly…

where is the graph of ( f(x)=4lfloor x - 3\rfloor+2 ) discontinuous?\nall real numbers\nall integers\nonly at multiples of 3\nonly at multiples of 4

where is the graph of ( f(x)=4lfloor x - 3\rfloor+2 ) discontinuous?\nall real numbers\nall integers\nonly at multiples of 3\nonly at multiples of 4

Answer

Explanation:

Step1: Recall the property of the greatest - integer function

The greatest - integer function (y = \lfloor t\rfloor) is discontinuous at all integer values of (t).

Step2: Find the transformation of the function

For the function (f(x)=4\lfloor x - 3\rfloor+2), let (t=x - 3). The function (y = f(x)) is a transformation of the greatest - integer function (y=\lfloor t\rfloor) (a vertical stretch by a factor of 4 and a vertical shift up by 2 units). The function (y=\lfloor t\rfloor) is discontinuous when (t = n), where (n\in\mathbb{Z}). Substituting (t=x - 3), we set (x-3=n), then (x=n + 3). Since (n) is an integer, (x) is an integer.

Answer:

all integers