Let f(x)=x−[x]1+x−[x],x ϵ R, where [ x] denotes the greatest integer func

Question -

Let $f (x) = \frac{x - [x]}{1 + x - [x]}, x ϵ R$ , where [ x] denotes the greatest integer function. Then, the range of f is

Options:

A .

(0, 1)

B .

[0, \frac{1}{2})

C .

[0, 1]

D .

[0, \frac{1}{2}]

Answer: Option B
:
B
The graph of

y = x - [x]

is as shown below

When x is an integer,

x - [x] = 0

Hence, f(x) = 0 when x is an integer

x \to [x]

as x tends to an integer.
Let X =

x - [x]

So,

f (x) = \frac{X}{1 + X}, X ϵ [0, 1)

X \to 1, \frac{X}{1 + X} \to \frac{1}{2}

Hence, the range of f(x) is

[0, \frac{1}{2})

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