Problem 29 - College Algebra 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 is all real numbers except the solutions to the equation \(x^{3}-5 x^{2}-4 x+20 = 0\).

Step 1: Identify the Denominator

Identify the denominator of the function, which is \(x^{3}-5 x^{2}-4 x+20\). Here, we need to find the x-values that make it equal to zero.

Step 2: Solve the equation

Solve \(x^{3}-5 x^{2}-4 x+20 = 0\) for x. Since it is a cubic equation, one can use polynomial division, factorization, or the cubic formula to solve the equation. In most cases, as this cubic equation doesn't have easy rational roots, numerical or approximation methods like Newton's method might be required.

Step 3: Find the Domain

Assuming the root or roots of the equation are a, b and c, the domain of the function will therefore be all real numbers except a, b and c. That is, \(x \neq a, b, c\). So the solution will be in the form of intervals such as \((-\u221e, a) \cup (a, b) \cup (b, c) \cup (c, \u221e)\).