Question

Find the range of the function $f\left( x \right) = [x] - x$ , where $[x]$ is the greatest integer function.

Answer 1

Hint : The term $[x]$ is the greatest integer function which means that if x has a decimal value then that decimal value is excluded from x to give $[x]$ so we write the value of $[x]$ as [x] = x - \left\\{ x \right\\} where \left\\{ x \right\\} is the fractional value of x. This fraction value lies between $0$ and $1$ .

Complete step-by-step answer :
Given to us, a function $f\left( x \right) = [x] - x$ where $[x]$ is the greatest integer function.
We can now write the value of $[x]$ as [x] = x - \left\\{ x \right\\} where \left\\{ x \right\\} is the fractional value of x.
By rearranging the terms in the above equation, we get \left[ x \right] - x = - \left\\{ x \right\\}
Now, we know that the fractional value \left\\{ x \right\\} lies between zero and one. Hence we can write its range as 0 \leqslant \left\\{ x \right\\} < 1
So now the range of - \left\\{ x \right\\} would be opposite of the range of \left\\{ x \right\\} and it can be written as 0 \geqslant - \left\\{ x \right\\} > - 1
This inequality can also be written as - 1 < \- \left\\{ x \right\\} \leqslant 0
We already have calculated the value of - \left\\{ x \right\\} to be $\left[ x \right] - x$ so let us substitute this value in the above inequality. So now this inequality becomes $- 1 < \left[ x \right] - x \leqslant 0$ and can also be written as $- 1 < f\left( x \right) \leqslant 0$
So the range of the given function is from $- 1$ to zero without including the $- 1$ value so we use open brackets to represent it. However this includes $0$ so we use a closed bracket to represent it.
Hence, the range of the given function $f\left( x \right)$ is written as $( - 1,0]$
So, the correct answer is “ $( - 1,0]$ ”.

Note : It is to be noted that if the range of a function lies between the values a, b and both a and b are included in the range then we represent it as $\left[ {a,b} \right]$ . If the value a is not included in the range then it is represented as $(a,b]$ and if b is not included then we represent it as $[a,b)$