Problem 30 - Precalculus Essentials

Question

Find the domain of each function. $$f(x)=\frac{7 x+2}{x^{3}-2 x^{2}-9 x+18}$$

Step-by-Step Solution

Answer

The domain of the function is all real numbers except 2, 3, and -3.

Step 1: Factorize the Denominator

Factorize the equation \(x^{3}-2 x^{2}-9 x+18 =0\). This can be rewritten as \(x^{2}(x-2) - 9(x-2) = 0\), and further simplified to \((x^{2}-9)(x-2) = 0\). The denominator further factorizes to \((x-3)(x+3)(x-2)=0\)

Step 2: Find the Zeros of the Denominator

Solving the equation \((x-3)(x+3)(x-2)=0\) for x gives three solutions: \(x=2\), \(x=3\) and \(x=-3\). These are the values for which the function is undefined.

Step 3: Exclude the Zeros from the Real Numbers

The function is defined for all real numbers except for the zeroes of the denominator. Therefore, the domain of the function is all real numbers except 2, 3 and -3.