Problem 4 - College Algebra

Question

Find the domain of each rational function. $$g(x)=\frac{2 x^{2}}{(x-2)(x+6)}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x)=\frac{2 x^{2}}{(x-2)(x+6)}\) is all real numbers except 2 and -6.

Step 1: Understanding the Function

The function given is \(g(x)=\frac{2 x^{2}}{(x-2)(x+6)}\). The domain of this function is all real numbers except those that make the denominator equal to zero.

Step 2: Complex Roots

Factor the denominator to identify values that make it equal to zero. The denominator has terms \(x-2\) and \(x+6\). Setting each equal to zero will give the complex roots.

Step 3: Determine the Complex roots

For \(x-2=0\), \(x = 2\). And for \(x+6=0\), \(x = -6\). These are the values that make the denominator equal to zero.

Step 4: State the Domain

The domain of the function thus excludes these two values. Therefore, the domain of \(g(x) = \frac{2 x^{2}}{(x-2)(x+6)}\) is \(x \in \mathbb{R}, x \neq 2, -6\).