Consider the function f(x) = sqrt(5 - x) / (x^2 - 9). What is the domain of f?
Show the answer and explanation
Correct answer
A. (-∞, 5] except x = -3 and x = 3
Principle or equation
The domain of a function defined by a formula is the set of all real numbers for which the expression is defined. For a square root, the radicand must be nonnegative. For a fraction, the denominator must be nonzero.
Why this answer is correct
The radicand 5 - x must be ≥ 0, so x ≤ 5. The denominator x^2 - 9 = (x - 3)(x + 3) must not be zero, so x ≠ 3 and x ≠ -3. Both -3 and 3 are ≤ 5, so they are excluded. Thus domain is (-∞, 5] except x = -3 and x = 3.
Example
For f(x) = sqrt(x) / (x - 1), domain is [0, 1) ∪ (1, ∞).
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