Question

Find the domain of the function f(x)= sqrt(5-3x) / x^2-64 and write your answer in interval notation. Domain: Help: Click here for help entering intervals. You have 3 attempt(s) remaining before you will receive a new version of this problem.

Answer 1

The domain of the given function, considering restrictions from the square root and the denominator, is (-infinity, -8) U (-8, 5/3] U (8, infinity).

Step-by-step explanation:

The domain of a function represents all possible x-values that make the function defined. Considering the function f(x) = sqrt(5-3x) / x^2-64, there are two restrictions we should take into account:

1. Inside the square root, the domain is determined by the inequality 5 - 3x >= 0, because you cannot take the square root of a negative number. Solving for x, we get x <= 5/3.
2. In the denominator, x^2 - 64 cannot be equal to zero because division by zero is undefined. Therefore, x cannot be -8 or 8 because these values would make the denominator zero.

Putting these together, the domain of the function, in interval notation, is -infinity < x <= 5/3 excluding -8 and 8. Therefore, the domain is (-infinity, -8) U (-8, 5/3] U (8, infinity).

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