Question

Find the domain of ${{\cos }^{-1}}\left[ X \right]$ where $\left[ X \right]$ is the greatest integer function.

Answer 1

Hint: Here we have to find the domain of the inverse function given. Firstly we will consider the range of the inverse trigonometry function as $\pi \le {{\cos }^{-1}}x\le 2\pi $ and find the value of $x$ . Then we will replace $x=\left[ X \right]$ and by using the greatest integer functions definition form two cases and get the value of $X$ . Finally we will take the intersection of the two values obtained in each case and get our desired answer.

Complete step by step answer:
We have to find the domain of ${{\cos }^{-1}}\left[ X \right]$ where $\left[ X \right]$ is the greatest integer function.
So here we have to get the value of $X$ .
Now as we know the range of ${{\cos }^{-1}}x$ is $\left[ \pi ,2\pi \right]$ so we can write it as follows,
$\pi \le {{\cos }^{-1}}x\le 2\pi $
Multiplying cosine with each term we get,
$\cos \pi \le \cos {{\cos }^{-1}}x\le \cos 2\pi $
By inverse function formula $\cos {{\cos }^{-1}}a=a$ and $\cos \pi =-1$, $\cos 2\pi =1$ using it above we get,
$-1\le x\le 1$
Now substituting $x=\left[ X \right]$ we get,
$-1\le \left[ X \right]\le 1$
The greatest integer function denoted by $\left[ X \right]$ takes a real number as input and gives the output as the nearest integer which is equal to or less than the input number.
We will take two case for the two values given as follows:
(i) Consider $\left[ X \right]\ge -1$
So this means that $X$ can be any real number from $-1$ to $\infty $
$\Rightarrow X\in \left[ -1,\infty \right)$
(ii) Consider $\left[ X \right]\le 1$
So this mean the value of $X$ can be any real number between $-\infty $ and $2$
$\Rightarrow X\in \left( -\infty ,2 \right)$
So taking the intersection of the values from both the cases we get,
$\Rightarrow X\in \left[ -1,\infty \right)\cap \left( -\infty ,2 \right)$
$\Rightarrow X\in \left[ -1,2 \right)$

Note:
 The greatest integer function definition should be clear in order to get the domain of the function. We have taken the lowest value in case (i) as $-1$ because if we deviate a little from $-1$ the value of the greatest integer function will become $-2$ which will contradict our answer. Similarly we cannot consider the highest value as $2$ in case (ii) as our greatest integer value will come as $2$ which is a contradiction for our condition.