Find domain and range of √1+√1-√1-x^2 show necessary steps
To find the domain and range of the given expression:
√1+√1-√1-x^2
We need to consider two things:
1. The expression under the square root should be non-negative.
2. The square root of the expression should be a real number.
Step 1: Check the domain
The first step is to check the domain of the expression under the square root. We have:
1 + 1 - √(1 - x^2)
Simplifying this expression, we get:
2 - √(1 - x^2)
For the expression under the square root to be non-negative, we need:
1 - x^2 ≥ 0
Solving this inequality, we get:
-1 ≤ x ≤ 1
Therefore, the domain of the given expression is:
-1 ≤ x ≤ 1
Step 2: Check the range
To find the range of the expression, we need to find the minimum and maximum values of the expression. Let y denote the value of the expression:
y = √1+√1-√1-x^2
We can simplify the expression as follows:
y = √2 - √(1 - x^2)
Since the range of the square root function is [0, ∞), we have:
0 ≤ √(1 - x^2) ≤ 1
Substituting this in the expression for y, we get:
√1 ≤ y ≤ √2
Therefore, the range of the given expression is:
√1 ≤ y ≤ √2
Final Answer:
Domain: -1 ≤ x ≤ 1
Range: √1 ≤ y ≤ √2
To find the domain and range of the expression √(1 + √(1 - √(1 - x^2))), we need to consider the restrictions on the values of x.
Step 1: Start with the innermost expression, 1 - x^2, under the square root. The square root of a negative number is undefined, so we need to ensure that the inside of the square root is non-negative.
1 - x^2 ≥ 0
Step 2: Solve the inequality.
Adding x^2 to both sides, we get:
1 ≥ x^2
Step 3: Take the square root of both sides.
√1 ≥ √x^2
Which simplifies to:
1 ≥ |x|
Step 4: Analyze the absolute value inequality.
Since we have an absolute value inequality, we can rewrite it as two separate inequalities, one for positive values of x and one for negative values of x.
For positive values:
1 ≥ x
For negative values:
1 ≥ -x
Step 5: Combine the two inequalities.
1 ≥ x and 1 ≥ -x
Taking the intersection of these two inequalities, we have:
-1 ≤ x ≤ 1
This is the domain of the function.
Step 6: Move to the outermost expression, √(1 + √(1 - √(1 - x^2))). To find the range, we consider the possible values of the expression for the values of x in the domain obtained in step 5.
We know that √(1 - x^2) is non-negative, so the innermost square root is always positive.
1 - √(1 - √(1 - x^2)) ≥ 0
√(1 - √(1 - x^2)) ≤ 1
Taking the outer square root of both sides:
1 - √(1 - x^2) ≤ 1
√(1 - x^2) ≥ 0
This tells us that the range of the function is [0, 1].