Compute the domain of the real-valued ntion f(x)=sqrt(1-sqrt(2-x). Thank you!
since the domain of √u is u>=0, we need
2-x >= 0
That is, x <= 2
Next, we also need 1-√(2-x) >= 0, so
√(2-x) <= 1
2-x <= 1 means x >= 1
-(2-x) <= 1 means x <= 3
So, we have a final domain of 1 <= x <= 2
The graph is at
http://www.wolframalpha.com/input/?i=%E2%88%9A%281-%E2%88%9A%282-x%29%29
You can see that f(x) has a zero imaginary part only in the interval [1,2]. That is the domain of f(x) for real values.
To determine the domain of the function f(x) = sqrt(1 - sqrt(2 - x)), we need to consider the values of x for which the function is valid and real.
First, let's consider the innermost square root, sqrt(2 - x). Since the square root of a negative number is undefined in the real numbers, we need 2 - x to be non-negative:
2 - x ≥ 0
Solving for x, we have:
x ≤ 2
Next, we look at the outer square root, sqrt(1 - sqrt(2 - x)). To ensure that this square root is real, we need the expression inside to be non-negative:
1 - sqrt(2 - x) ≥ 0
To solve this inequality, we isolate the square root:
sqrt(2 - x) ≤ 1
Now, we square both sides of the inequality:
2 - x ≤ 1
Solving for x, we get:
x ≥ 1
Combining this with our previous result, we find that the function is defined for:
1 ≤ x ≤ 2
Therefore, the domain of the function f(x) = sqrt(1 - sqrt(2 - x)) is [1, 2].
To compute the domain of the function f(x) = sqrt(1 - sqrt(2 - x)), we need to consider two things:
1. The radicand inside each square root must be non-negative.
2. The denominator inside the square root must not be zero.
Let's start with the first condition:
1. The radicand inside the outer square root, 1 - sqrt(2 - x), must be non-negative.
To satisfy this condition, we set 1 - sqrt(2 - x) ≥ 0 and solve for x:
1 - sqrt(2 - x) ≥ 0
sqrt(2 - x) ≤ 1
2 - x ≤ 1 (Squaring both sides, remembering to flip the inequality since we squared a negative square root)
2 - 1 ≤ x
1 ≤ x
So, the domain of the function is x ≥ 1.
Now let's move to the second condition:
2. The denominator inside the inner square root, 2 - x, must not be zero.
To satisfy this condition, we set 2 - x ≠ 0 and solve for x:
2 - x ≠ 0
x ≠ 2
So, the domain of the function also excludes x = 2.
Combining both conditions, the domain of the function f(x) = sqrt(1 - sqrt(2 - x)) is x ≥ 1, excluding x = 2.