Question

The range of $f(x)=[\cos x]$, where $[x]$ is the greatest integer less than or equal to $x$, is

• A. (-1,1)
• B. \{-1,0\}
• C. \{1,0\}
• D. \{-1,0,1\}
• E. [-1,1]

Hint:

Logic Tip: Brackets like $\{\}$ denote a set of discrete values, whereas parentheses $()$ and square brackets $[]$ denote continuous intervals. Since the greatest integer function strictly outputs integers, the answer must be a discrete set, eliminating options A and E immediately.

Answer 1

The Correct Option is D

Concept:
The range of a function represents all possible output values. The greatest integer function (or floor function) $[y]$ outputs the largest integer that is less than or equal to $y$. Therefore, the output of $[y]$ will always be an integer.
Step 1: Determine the range of the inner function.
The inner function is $\cos x$. The standard range for the cosine function for all real numbers $x$ is: $$-1 \le \cos x \le 1$$
Step 2: Apply the greatest integer function to the bounds.
We need to evaluate $[\cos x]$ for values in the interval $[-1, 1]$.

• When $\cos x = 1$ (e.g., at $x = 0$), $[\cos x] = [1] = 1$.
• When $0 \le \cos x<1$ (e.g., at $x = \pi/3$), $[\cos x] = [0.5] = 0$.
• When $-1 \le \cos x<0$ (e.g., at $x = 2\pi/3$), $[\cos x] = [-0.5] = -1$.

Step 3: Compile the final range set.
Since the greatest integer function strictly outputs integers, the only possible outputs produced by the interval $[-1, 1]$ are the discrete integers contained within and at the boundaries of that interval. Thus, the range is the set of these distinct values: $$\{-1, 0, 1\}$$