Where is the graph of [tex]f(x)=4\lfloor x-3\rfloor+2[/tex] discontinuous?
A. all real numbers
B. all integers
C. only at multiples of 3
D. only at multiples of 4
Answer (1)
The floor function ⌊ x ⌋ is discontinuous at all integers.
The function ⌊ x − 3 ⌋ is discontinuous when x − 3 is an integer, which means x is an integer.
The function f ( x ) = 4 ⌊ x − 3 ⌋ + 2 is a transformation of the floor function, and its discontinuities occur at the same points as ⌊ x − 3 ⌋ .
Therefore, the graph of f ( x ) is discontinuous at all integers. all integers
Explanation
Understanding the Problem The problem asks us to identify the points of discontinuity of the function f ( x ) = 4 ⌊ x − 3 ⌋ + 2 . The key here is understanding the behavior of the floor function.
Discontinuity of the Floor Function The floor function, denoted by ⌊ x ⌋ , returns the greatest integer less than or equal to x . This function is discontinuous at every integer value. That is, at every integer n , the limit from the left, lim x → n − ⌊ x ⌋ = n − 1 , is not equal to the limit from the right, lim x → n + ⌊ x ⌋ = n .
Discontinuity of ⌊ x − 3 ⌋ Now, consider the function ⌊ x − 3 ⌋ . This is simply a horizontal shift of the standard floor function. The points of discontinuity will occur when x − 3 is an integer. Let x − 3 = n , where n is an integer. Then, x = n + 3 . Since n is an integer, n + 3 is also an integer. Therefore, ⌊ x − 3 ⌋ is discontinuous at all integers.
Discontinuity of f ( x ) Finally, consider the function f ( x ) = 4 ⌊ x − 3 ⌋ + 2 . This is a vertical stretch by a factor of 4 and a vertical shift by 2 of the function ⌊ x − 3 ⌋ . Vertical stretches and shifts do not change the points of discontinuity. Therefore, f ( x ) is discontinuous at all integers.
Examples
The floor function and its discontinuities are crucial in various real-world applications, such as digital signal processing, computer graphics, and control systems. For instance, in manufacturing, if you need to count the number of items that fit into a box, and the number of items, x , can be a real number, then ⌊ x ⌋ gives you the number of whole items that fit. Understanding where this function jumps is essential for accurate calculations and avoiding errors in automated systems.
