To determine the intervals where the function f(x) = 2 + √(-16 - 2x) is continuous, we need to first understand the nature of the square root function. The square root function is defined only for non-negative numbers. That is, the expression under the square root (√) must be greater than or equal to zero (≥0).
If we consider the function f(x) = 2 + √(-16 - 2x), the expression within the square root is -16 - 2x.
For a square root to be defined in the real number system, the value within the root must be non-negative. Therefore, we set -16 - 2x ≥ 0, i.e., -2x ≥ 16.
However, this inequality gives us no real solutions. We can see that the inequality -2x ≥ 16 would imply that x ≤ -8. However, if we substitute a real x-value ≤ -8, the term under the square root, -16 - 2x, remains negative due to the negative 16 at the beginning of the expression. As a result, there is no real x that makes the original equation non-negative. Therefore, the function f(x) = 2 + √(-16 - 2x) is not defined in the Real domain.
Because the function is not defined for any real values of x, it is also not continuous on the real line. A function must be defined at a point to be continuous at that point, but since there are no points where this function is defined, there are also no points where it is continuous. Thus, we say the set of x where the function is continuous is the Empty Set, often denoted as ∅.
Therefore, the function f(x) = 2 + √(-16 - 2x) is not continuous on any interval of real numbers.