What is the domain of the following function? $$f(x) = \sqrt {\frac {19 x}{x^{2} - 36}}$$

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The problem asks for the domain of f(x) = sqrt(19x / (x^2 - 36)).
First, remember that a square root requires its radicand to be nonnegative and the denominator of the fraction cannot be zero.
Identify the critical points where the expression could change sign or be undefined: x = -6, x = 0, and x = 6.
We will test the sign of the fraction 19x / (x^2 - 36) across the intervals determined by these points: (-∞, -6), (-6, 0), (0, 6), (6, ∞).
Interval (-∞, -6): choose x = -7. Numerator 19x is negative; denominator x^2 - 36 = 49 - 36 = 13 is positive. Negative over positive is negative, so not allowed.
Interval (-6, 0): choose x = -1. Numerator 19x is negative; denominator (-1)^2 - 36 = 1 - 36 = -35 is negative. Negative over negative is positive, so this interval is allowed. At x = -6 the denominator is zero (not allowed), so -6 is excluded; at x = 0 the numerator is zero, giving zero inside the square root, which is allowed, so 0 is included. Thus this interval contributes (-6, 0].
Interval (0, 6): choose x = 1. Numerator positive; denominator 1^2 - 36 = -35 is negative. Positive over negative is negative, so not allowed.
Interval (6, ∞): choose x = 7. Numerator positive; denominator 49 - 36 = 13 positive. Positive over positive is positive, so this interval is allowed; 6 itself is excluded due to division by zero at x = 6.
Combining the allowed intervals and including the endpoint 0, the domain is (-6, 0] ∪ (6, ∞).
Note: The prompt’s answer options aren’t provided here, but the derived domain matches the standard domain given by ensuring the radicand is nonnegative and the denominator is nonzero.

MATH1061/1002/1021 (ND) MATH1061 Canvas Quiz 10

What is the domain of the following function? \[f(x) = \sqrt {\frac {19 x}{x^{2} - 36}}\]

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