The function `f(x) = sqrt(x - sqrt(1 - x^2))`
The domain of a function f(x) is the set of values that x can take and for which f(x) is real and defined.
For the function `f(x) = sqrt(x - sqrt(1 - x^2))` , as the square root of negative numbers is not defined, `1 - x^2 > 0 `
=> `1 > x^2`
=> `-1 < x < 1` ...(1)
and `x - sqrt(1 - x^2) > 0`
As x^2 is a positive value, x cannot be negative.
=> `x > sqrt(1 - x^2)`
=> `x^2 > 1 - x^2`
=> `2x^2 > 1`
=> `x^2 > 1/2`
=> `x > 1/sqrt 2` ...(2)
The domain of the function is `(1/sqrt 2, 1)`
The range of the function is all the values it can take for x lying in the domain. The range of f(x) is [0, 1]
The domain of the given function is `(1/sqrt 2, 1)` and the range is `[0, 1]`