Question

Find the domain of the function. $$ g(x)=\frac{1}{\left|x^{2}-4\right|} $$

Short answer

The domain of the function \( g(x)=\frac{1}{\left|x^{2}-4\right|} \) is \( (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \).

Step-by-step solution

Identify the denominator

The denominator of the function is \( |x^{2}-4| \). The function is defined for all values of x except those that make the denominator equal to zero.

Solve the equation |x²-4|=0 for x

To determine the values of x that make the denominator zero, we solve the equation \( |x^{2}-4|=0 \). The absolute value of a number is zero if and only if the number itself is zero. Hence, this equation reduces to \( x^{2}-4=0 \). Solving this equation yields two solutions: \( x=-2, 2 \). That means when \( x = -2 \) or \( x = 2 \), the function is undefined.

Establish the domain

So, the function \( g(x)=\frac{1}{\left|x^{2}-4\right|} \) is defined for all real numbers except for \( x=-2 \) and \( x=2 \). Therefore, the domain of the function is \( (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \). This notation means all real numbers excluding -2 and 2.