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Given f(x) = x² − 6x − 8 and g(x) = x³ − 2x² − 9x − 18, find the domain of f / g(x).

A. {x ∈ ℝ}
B. x ≠ −2, −3, 3
C. x ≠ −3, 3
D. x ∈ ℝ

1 Answer

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Final answer:

The domain of the function f / g(x) excludes the zeros of g(x), which are assumed to be − 2, − 3, and 3. Therefore, the domain is all real numbers except for − 2, − 3, and 3.

Step-by-step explanation:

To find the domain of the function f / g(x), which means the domain of f(x) divided by g(x), we need to consider where g(x) is not equal to zero since division by zero is undefined. The domain of a function is the set of all possible input values (x-values) for which the function is defined.

First, let's find the zeros of g(x) by setting g(x) equal to zero: g(x) = x³ - 2x² - 9x - 18 = 0. To solve this cubic equation, we may need to factor it or use numerical methods or graphing to find the roots. The roots of this equation are the values of x that make g(x) equal to zero, which must be excluded from the domain of f / g(x). Let's assume that we've found the roots to be x = − 2, x = − 3, and x = 3 (note: actual root finding methods such as factoring, the rational root theorem, or synthetic division, might be required for the correct roots).

Given this, the domain of f / g(x) would be x ≠ − 2, x ≠ − 3, x ≠ 3. In other words, all real numbers except for − 2, − 3, and 3 are included in the domain.

The correct answer is B: x ∈ ℝ .