What is the largest possible domain for the function

f(x) = ~x + ~(4 - x^2) ?
a.{x: O:::;x:::;2,x ER}
b.{x: -2 :::;x :::;2, x E R}
c.{x: 0 s X :s 4, X E:R}
d.{x: x 2': 2, x ER}

To determine the largest possible domain for the function f(x) = ~x + ~(4 - x^2), we need to consider the restrictions or values that x can take while still maintaining a valid function.

Let's analyze each option:

a. {x: O:::;x:::;2,x ER}
This option specifies that x should be greater than or equal to 0 and less than or equal to 2. However, the function involves taking the square of x (x^2), so there is no restriction on x being greater than 2 or less than 0. Therefore, this option is not the largest possible domain.

b. {x: -2 :::;x :::;2, x E R}
This option restricts x to be between -2 and 2, inclusive. Since the function does not have any restrictions based on the range of x values, this option is valid. However, we need to determine if it is the largest possible domain.

c. {x: 0 s X :s 4, X E:R}
This option states that x should be between 0 and 4, inclusive. Similar to option a, the function involves taking the square of x, so there is no restriction on x being greater than 4. Thus, this option is not the largest possible domain.

d. {x: x 2': 2, x ER}
This option is not clear due to the formatting of the question. It appears to say "x 2': 2," which does not make sense. Therefore, we cannot consider this option as a valid choice.

After analyzing the given options, we find that option b. {x: -2 :::;x :::;2, x E R} is the largest possible domain for the function f(x) = ~x + ~(4 - x^2). It allows x to take any real value between -2 and 2, inclusive, without any additional restrictions.