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

Hint: 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.