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Determine the vertical asymptote of the following rational function: f(x) = (x - 3) / (x + 2) a. x = -2 b....

1. Determine the vertical asymptote of the following rational function: f(x) = (x - 3) / (x + 2)
a. x = -2
b. y = 1
c. y = -2
d. x = 1
2. Determine the domain of the following rational function: f(x) = (3x + 2) / (x - 5)
a. (-∞, 3) ∪ (3, ∞)
b. (-∞, -3) ∪ (-3, ∞)
c. (-∞, 5) ∪ (5, ∞)
d. (-∞, -5) ∪ (-5, ∞)

Answer

  1. The vertical asymptote of a rational function occurs where the denominator is zero and the numerator is not zero. For f(x) = (x - 3) / (x + 2), the denominator x + 2 = 0 when x = -2. Therefore, the vertical asymptote is at x = -2.

  2. The domain of a rational function is all real numbers except where the denominator is zero. For f(x) = (3x + 2) / (x - 5), the denominator x - 5 = 0 when x = 5. Therefore, the domain is all real numbers except x = 5, which is (-∞, 5) ∪ (5, ∞).

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