What is the range and domain of `f(x) = sqrt(x - sqrt(1 - x^2))`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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]`

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