Final answer:
The domain of the function f(x) = (x-6) / (x^2-3x-18) is (-∞,-3) ∪ (-3, 6) ∪ (6, ∞) and the range is (-∞,0) ∪ {0} ∪ (0, ∞).
Step-by-step explanation:
The function f(x) is given as f(x) = (x-6) / (x^2-3x-18). To determine the domain and range, we need to consider any restrictions on the values of x. In this case, since the function is defined for all real numbers except for those that make the denominator zero, we need to find the values of x that satisfy the equation x^2-3x-18 = 0. Solving this quadratic equation, we find that x = -3 and x = 6. Therefore, the domain of f is (-∞,-3) ∪ (-3, 6) ∪ (6, ∞).
To find the range of f, we can analyze the behavior of the function as x approaches positive and negative infinity. As x approaches negative infinity, the value of f(x) approaches 0. As x approaches positive infinity, the value of f(x) also approaches 0. Therefore, the range of f is (-∞,0) ∪ {0} ∪ (0, ∞).