Problem 29 - Precalculus Essentials

Question

Find the domain of each function. $$f(x)=\frac{2 x+7}{x^{3}-5 x^{2}-4 x+20}$$

Step-by-Step Solution

Answer

The domain of the function \( f(x) = \frac{2x + 7}{x^{3} - 5x^{2} - 4x + 20} \) is all real numbers except the 'x' values that make the denominator equal to zero.

Step 1: Set the Denominator Equal to Zero

To find the 'x' values that make the function undefined, set the denominator equal to zero and solve for 'x': \(x^{3} - 5x^{2} - 4x + 20 = 0\)

Step 2: Solve for x

Solving a cubic equation can be challenging. First, try factoring or use synthetic division to find the roots. In case it's not possible to factorize easily, use numerical methods or graphical solutions or use the cubic formula to solve for x.

Step 3: Determine the Domain

After finding the 'x' values that make the denominator zero, exclude these values from the domain. The set of all other real numbers will be the domain of the function.