Final answer:
The domain of the function f(x)=√(x^3-16x) is (-∞, -4] U (0, 4] and the least value in the domain is -4.
Step-by-step explanation:
To find the domain of the function f(x)=√(x^3-16x), we need to consider the values of x that would make the expression inside the square root non-negative. This is because the square root function is only defined for non-negative numbers.
So, we set the expression inside the square root greater than or equal to zero:
x^3-16x ≥ 0
To solve this inequality, we can factor out x:
x(x^2-16) ≥ 0
Then, we find the critical points by setting each factor equal to zero:
x = 0 and x^2-16 = 0
From the second equation, we get x = -4 or x = 4. So, the critical points are x = 0, x = -4, and x = 4.
We can now create a sign chart to determine the intervals where the expression is positive or negative. Checking the values of x in each interval will help us find the domain.
From the sign chart, we see that the function is positive for x < -4 or 0 < x < 4. Therefore, the domain of the function is (-∞, -4] U (0, 4].
The least value in the domain is -4.