Final answer:
The domain of the function f(x) = (x - 6) / (x² - 3x - 18) is (-∞, -3) ∪ (-3, 6) ∪ (6, ∞), and the range is (-∞, ∞), as x = 6 and x = -3 cause the function to be undefined and the function's outputs can take any real value.
Step-by-step explanation:
The domain and range of a function are all the possible inputs (x-values) and outputs (y-values) respectively. For the function f(x) = \frac{x - 6}{{x^2 - 3x - 18}}, the denominator factors to (x-6)(x+3), so the function has discontinuities (where the function is not defined) at x = 6 and x = -3 because division by zero is undefined. Therefore, the domain of f(x) is all real numbers except x = 6 and x = -3, which is mathematically expressed as (-∞, -3) ∪ (-3, 6) ∪ (6, ∞).
As for the range, we need to look at the behavior of the function as x approaches the values that cause the function to be undefined and consider the horizontal asymptotes or extreme values of the function to determine the set of possible outputs. Since there's no explicit constraint on y-values and the function doesn't have a horizontal asymptote as f(x) approaches ±∞ (because the degree of the polynomial in the numerator is one less than that in the denominator), the y-values can take on any real number as x approaches the excluded points from the left and right. Thus, the range of f(x) is all real numbers, or (-∞, ∞).