What is the largest possible domain for the function?
F(x) = �ã x + �ã (4 - x^2)
To find the largest possible domain for the function F(x) = √(x + √(4 - x^2)), we need to consider the restrictions on x based on the properties of the square root function.
The square root function (√) is defined for non-negative real numbers. Therefore, the expression inside the first square root, x + √(4 - x^2), must be greater than or equal to 0, in order for the function to be defined.
First, let's consider the expression inside the second square root, 4 - x^2. This expression represents the radicand (the value inside the square root) and must be non-negative.
To solve the inequality 4 - x^2 ≥ 0, we need to identify the values of x that make the expression greater than or equal to 0. We can factor the expression as (2 - x)(2 + x) and set each factor to zero:
(2 - x)(2 + x) ≥ 0
When we solve (2 - x) = 0, we find x = 2.
When we solve (2 + x) = 0, we find x = -2.
These values indicate that the expression (2 - x)(2 + x) changes sign at x = -2 and x = 2.
Therefore, to satisfy the inequality (4 - x^2 ≥ 0), we consider the following cases:
1. When x < -2: In this case, both factors (2 - x) and (2 + x) are negative, meaning that their product is positive. Thus, the inequality is satisfied.
2. When -2 ≤ x ≤ 2: In this case, the factor (2 - x) is positive, while the factor (2 + x) is negative. This results in a negative product, violating the inequality.
3. When x > 2: In this case, both factors (2 - x) and (2 + x) are positive, meaning their product is also positive. Thus, the inequality is satisfied.
From the analysis above, we can see that the inequality (4 - x^2) ≥ 0 is satisfied for x ≤ -2 and x ≥ 2.
Now, let's consider the expression inside the first square root, x + √(4 - x^2). As mentioned earlier, this expression must be greater than or equal to 0.
Combining the restriction from the square root with the range of values for x obtained from solving the inequality, we find that the largest possible domain for the function F(x) = √(x + √(4 - x^2)) is:
x ≤ -2 and x ≥ 2.