Problem 30 - College Algebra 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 \(f(x)=\frac{7 x+2}{x^{3}-2 x^{2}-9 x+18}\) is all real numbers except x=1, x=3, and x=-6.

Step 1: Set up the equation

To find the value of x that makes the denominator zero, set the denominator equal to zero and solve for x. The equation would look like this: \(x^{3}-2 x^{2}-9 x+18=0\).

Step 2: Solving the equation

Solving the cubic equation can be tricky, in this case, factorizing the expression might be a useful method. Factoring the cubic equation \(x^{3}-2 x^{2}-9 x+18\) yields \((x-1)(x^2-x-18)\). Now, these could further be factored into simpler expressions particularly the second term as \((x-1)(x-3)(x+6)\). By setting the individual factors equal to zero, we get the roots of the equation as x=1, x=3, x=-6.

Step 3: Excluding the undefined points from the domain

The roots of the equation indicate the values of x for which the function is undefined. Therefore, to specify the domain of the function, we need to exclude these points. Though x could be any real number, it can't be 1, 3, or -6 as these would make the denominator zero and the function undefined.