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Chapter 3: Problem 4

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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 \(x \in \mathbb{R}, x \neq 2, -6\).

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