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) $