Question

The range of $f\left( x \right) = \cos \left[ x \right]$ , $- \dfrac{\pi }{4} < x < \dfrac{\pi }{4}$ where $\left[ x \right]$ represents greatest integer function less than or equal to $x$ is
A. \left\\{ 0 \right\\}
B. $\left[ { - 1,1} \right]$
C. \left\\{ {\cos 1,1} \right\\}
D. \left\\{ { - 1,1} \right\\}

Answer 1

In the above given problem, we are given a function $f\left( x \right)$ as $f\left( x \right) = \cos \left[ x \right]$ , for the interval $- \dfrac{\pi }{4} < x < \dfrac{\pi }{4}$ . Here $\left[ x \right]$ represents greatest integer function less than or equal to $x$ . We have to find the range of the function $f\left( x \right)$ . In order to approach the solution, first we have to rewrite the interval for the values of $x$ . After that we can change the obtained interval for $x$ to the interval for $\left[ x \right]$ . After that, in the end, we can change the interval again for the function $f\left( x \right)$ and then we can write the range for the given function $f\left( x \right)$.

Complete step by step answer:
Given that, the cosine function in composition with the greatest integer function written as,
$\Rightarrow f\left( x \right) = \cos \left[ x \right]$
Where the interval for the values of $x$ is given as $- \dfrac{\pi }{4} < x < \dfrac{\pi }{4}$ .
Now, we can rewrite the interval for $x$ by writing the values in decimals.
Since $\dfrac{\pi }{4} = 0.785$ , therefore the new interval for the values of $x$ can be written as,
$\Rightarrow - 0.785 < x < 0.785$
Now for the greatest integer function $\left[ x \right]$ , we have $\left[ { - 0.785} \right] = - 1$ and $\left[ {0.785} \right] = 0$ .

Therefore, the interval for the greatest integer function $\left[ x \right]$ can be written as,
$\Rightarrow - 1 < \left[ x \right] < 0$
Now, similarly for the cosine function, we have $\cos \left( { - 1} \right) = \cos 1$ and $\cos \left( 0 \right) = 1$ .
Therefore, the interval for the cosine function $\cos \left[ x \right]$ can be written as,
$\Rightarrow \cos 1 < \cos \left[ x \right] < 1$
That is the required interval for the function $f\left( x \right) = \cos \left[ x \right]$ .
Therefore, the range for the function $f\left( x \right) = \cos \left[ x \right]$ is \left\\{ {\cos 1,1} \right\\}.

Hence, the correct option is C.

Note: The greatest integer function is a function that gives the greatest integer less than or equal to the operated number. The greatest integer less than or equal to a number $x$ is represented as $\left[ x \right]$ . We have to round off the given number to the nearest integer that is less than or equal to the number itself. For example, $\left[ {0.2} \right] = 0$ , $\left[ {1.2} \right] = 1$ , $\left[ \pi \right] = 3$ , $\left[ e \right] = 2$ , $\left[ {\sqrt 3 } \right] = 1$ , $\left[ { - 0.01} \right] = - 1$ , $\left[ {3.99} \right] = 3$ , $\left[ { - 2.01} \right] = - 3$ , etc.