Problem 10 - Precalculus Essentials

Question

Find the domain of each function. $$f(x)=\frac{1}{x+8}+\frac{3}{x-10}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \frac{1}{x + 8} + \frac{3}{x - 10}\) is all real numbers except -8 and 10.

Step 1: Set the Denominators Equivalent to Zero

Setting the denominator of the first fraction to zero, we obtain \(x + 8 = 0\). Solving for x gives \(x = -8\). For the second fraction, setting the denominator equivalent to zero, \(x - 10 = 0\) gives \(x = 10\) as the solution.

Step 2: Formulate the Domain

The domain of the function is all real numbers except the solutions obtained in the previous step, because at these points, the function becomes undefined. Therefore, the domain of the function \(f(x) = \frac{1}{x + 8} + \frac{3}{x - 10}\) is all real numbers except -8 and 10.