Question

The range of f(x) = $\frac{x-\left[x\right]}{1+x-\left[x\right]}$ where [] represents greatest integer function.

• A. [0,1]
• B. [0, ½]
• C. [0, ½)
• D. None of these.

Answer 1

The correct option is C

[0, ½)

The given function can also be written as –

f(x) = $\frac{\left\{x\right\}}{1+\left\{x\right\}}$

Here, Let’s first find the domain of f(x). We know that {x} gives values from [0,1) for all real values of x. So we can put any value of x and the function will be defined. So the domain of f(x) would be R.

Now, f(x) = $\frac{\left\{x\right\}}{1+\left\{x\right\}}$

Now, in order to find the range let’s write the above equation as x = g(y)

{x} = $\frac{y}{1-y}$

We know that {x} lies from [0,1)

$0\le$ {x} < 1\)

Or

$0\le \frac{y}{1-y}<1$

Or $yϵ[0,\frac{1}{2})$