Answer:
To find the domain of the function p(x) = √(-30 - 11x - x^2), we need to determine the values of x for which the function is defined.
The function p(x) is defined if the expression under the square root (√) is greater than or equal to zero, since the square root of a negative number is not a real number. So we have:
-30 - 11x - x^2 ≥ 0
To solve this inequality, we can factor the quadratic expression:
-(x^2 + 11x + 30) ≥ 0
Now, let's factor the quadratic:
-(x + 6)(x + 5) ≥ 0
To find the values of x that satisfy the inequality, we can analyze the signs of the factors and their product. We have three intervals to consider:
Interval 1: x < -6
In this interval, both (x + 6) and (x + 5) are negative, so their product is positive. Therefore, the inequality -(x + 6)(x + 5) ≥ 0 is satisfied in this interval.
Interval 2: -6 < x < -5
In this interval, (x + 6) is positive, and (x + 5) is negative. Their product is negative, so the inequality is not satisfied in this interval.
Interval 3: x > -5
In this interval, both (x + 6) and (x + 5) are positive, so their product is positive. The inequality -(x + 6)(x + 5) ≥ 0 is satisfied in this interval as well.
Therefore, the domain of the function p(x) is the union of intervals 1 and 3, which can be written in interval notation as:
(-∞, -6] ∪ (-5, +∞)