What is the domain of #f(x)=sqrt{1-sqrt[1-sqrt(1-x^2)]}# ?

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Answer 1 · 2017-12-15T20:07:05

The domain of #f(x)# is #[-1, 1]# and its range is #[0, 1]#

Given:

#f(x) = sqrt(1-sqrt(1-sqrt(1-x^2)))#

Note that the square of any real number is non-negative. So #sqrt(t)# only has real values for #t >= 0#

So #sqrt(1-x^2)# is only defined as a real valued square root if:

#1-x^2 >= 0#

Note that if #x# is real then #x^2 >= 0#. Hence the range of values for which #1-x^2 >= 0# is #x in [-1, 1]#.

Suppose #x in [-1, 1]#. Then:

So the domain of #f(x)# is #[-1, 1]# and its range is #[0, 1]#