What is the range of f(x) = (x)/sqrt (x-1)

The range of a function is the set of the values of f(x)
when x lies in the domain.

The domain of f(x) is all values
where x – 1 > 0 because if the denominator is 0, we get an indeterminate number
and for x - 1 < 0, the square root is a complex
number.

=> x >
1

f(x) = x/sqrt(x - 1)

f'(x) =
[1*sqrt(x - 1) - x*(1/2)(1/sqrt(x - 1))]/(x - 1)

=>
f'(x) = [(x - 1) - x/2]/(x - 1)^(3/2)

=> f'(x) = (x
- 2)/2*(x - 1)^(3/2)

equating f'(x) = 0, we get x =
2

The function has the minimum value when x =
2

f(2) = 2/sqrt 1 = 2

If x
> 1, f(x) >= 2

The range of the
function is (inf., 2]