Consider the function f(x) = sqrt(6 - x) / (x^2 - 7x + 10). What is the domain of f?
Show the answer and explanation
Correct answer
A. (-∞, 2) ∪ (2, 5) ∪ (5, 6]
Principle or equation
The domain of a function defined by a fraction with a square root in the numerator requires the radicand to be nonnegative and the denominator to be nonzero.
Why this answer is correct
The radicand 6 - x must be ≥ 0, so x ≤ 6. The denominator x^2 - 7x + 10 = (x - 2)(x - 5) must not be zero, so x ≠ 2 and x ≠ 5. Combining these gives (-∞, 2) ∪ (2, 5) ∪ (5, 6].
Example
For f(x) = sqrt(4 - x)/(x - 1), domain is (-∞, 1) ∪ (1, 4].
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