Compute the domain of the real-valued function f(x)=sqrt(1-sqrt(2-x)). Thank you

Answer 1

first of all,
2-x > 0
-x > -2
x < 2

furthermore,
1-√(2-x) > 0
1 > √(2-x)
√(2-x) < 1
so 0 < 2-x < 1
-2 < -x < -1
2 > x > 1
or 1 < x < 2

so if x < 2 AND 1 < x < 2
then

1 < x < 2

test:
take a value outside that domain:
let x = 5
f(5) = √( 1 - √-3)) , not defined for the reals
let x = .6
f(.6) = √(1 - √1.4)
= √(-.1832...) which is undefined as well

but x = 1.3
f(1.3) = √(1 - √.7)
= √.16333997...) = a real number