Find the domain of the function. Write in interval notation.
sqrt(x^2 − 64).
For the answer, I got (-infinity, -8)u(8, infinity), but that's wrong, and I'm not sure what else to try?
if you allow taking imaginary roots, then the domain is -inf to +inf.
If you allow only real roots, then must be less than -8 or greater than 8, your answer.
consider x=3
sqrt(9-64)=sqrt(-55)=i*sqrt55
To find the domain of the function sqrt(x^2 - 64), we need to consider the restrictions on the input value, x, that would make the function undefined.
The square root function is defined for all non-negative real numbers. However, we also need to ensure that the expression inside the square root is non-negative.
In this case, we have sqrt(x^2 - 64). For this expression to be non-negative, we need x^2 - 64 to be greater than or equal to zero.
Let's solve this inequality:
x^2 - 64 ≥ 0
Factor the left side:
(x - 8)(x + 8) ≥ 0
Now, we can identify the critical points by setting each factor equal to zero:
x - 8 = 0 → x = 8
x + 8 = 0 → x = -8
These critical points divide the number line into three intervals: (-infinity, -8), (-8, 8), and (8, infinity).
Now, we can test a value from each interval to determine whether the expression is positive or negative:
- Choose x = -9 from (-infinity, -8):
Substituting this value into the inequality, we get (-9 - 8)(-9 + 8) ≥ 0 → (-17)(-1) ≥ 0 → 17 ≥ 0. This interval satisfies the inequality.
- Choose x = 0 from (-8, 8):
Substituting this into the inequality, we get (0 - 8)(0 + 8) ≥ 0 → (-8)(8) ≥ 0 → -64 ≥ 0. This interval does not satisfy the inequality.
- Choose x = 9 from (8, infinity):
Substituting this into the inequality, we get (9 - 8)(9 + 8) ≥ 0 → (1)(17) ≥ 0 → 17 ≥ 0. This interval satisfies the inequality.
From this analysis, we can conclude that the domain of the function sqrt(x^2 - 64) is (-infinity, -8) ∪ (8, infinity).
Therefore, the correct answer in interval notation is (-∞, -8) ∪ (8, ∞).