how do you find the domain of g(x)= x^2-4 / 2x-x^2

To find the domain of the function \( g(x) = \frac{x^2 - 4}{2x - x^2} \), you need to determine where the function is defined. Specifically, you want to identify the values of \( x \) for which the denominator is non-zero since division by zero is undefined.

### Steps to find the domain:

1. **Denominator Condition:**
   The function is undefined when the denominator is zero. So, we need to solve for when:
   \[
   2x - x^2 = 0
   \]
   Factor the quadratic expression:
   \[
   x(2 - x) = 0
   \]
   This gives two solutions:
   \[
   x = 0 \quad \text{or} \quad x = 2
   \]
   These are the values of \( x \) that would make the denominator zero, so the function is undefined at \( x = 0 \) and \( x = 2 \).

2. **Domain:**
   The domain of the function will exclude these values. Therefore, the domain is:
   \[
   x \in (-\infty, 0) \cup (0, 2) \cup (2, \infty)
   \]
   This represents all real numbers except \( x = 0 \) and \( x = 2 \).

Would you like further details or any clarifications? Here are some follow-up questions you might consider:

1. What is the range of this function?
2. How do vertical asymptotes relate to the domain restrictions?
3. Can you graph this function to visualize the behavior near the undefined points?
4. How would the function behave as \( x \to \infty \) or \( x \to -\infty \)?
5. How can you express this function in factored form?

**Tip:** Always check both the numerator and denominator for possible restrictions when finding the domain of rational functions.

how do you find the domain of g(x)= x^2 - 4 / 2x - x^2

Learn how to find the domain of the rational function g(x) = (x^2 - 4) / (2x - x^2). This detailed solution includes factoring the quadratic, solving for when the denominator is zero, and identifying domain restrictions. Ideal for students in Grades 9-11 studying algebra and rational functions.

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