Problem 7 - College Algebra Essentials

Question

Find the domain of each function. $$g(x)=\frac{3}{x^{2}-2 x-15}$$

Step-by-Step Solution

Answer

The domain of the function g(x) is \(-\infty, -3\) union \(-3, 5\) union \(5, \infty\)

Step 1: Set the Denominator Equal to Zero and Solve

The denominator of the function is \(x^{2}-2 x-15\). Setting this equal to zero will give the equation \(x^{2}-2 x-15 = 0\). This can be factored to give \((x-5)(x+3) = 0\). Setting each factor equal to zero gives the solutions x = 5 and x = -3.

Step 2: Determine the Domain of the Function

The domain is all real numbers except for the values that make the denominator zero, which are x = 5 and x = -3. So the domain of the function g(x) is \(-\infty, -3\) union \(-3, 5\) union \(5, \infty\). This represents all real numbers except x = -3 and x = 5.