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Say whether it is even, odd or neither. Give reasons

To determine whether the function \( g(x) = x^4 + 3x^2 - 1 \) is even, odd, or neither, we can follow these steps:

### 1. **Check for Evenness:**
   A function \( g(x) \) is **even** if \( g(-x) = g(x) \) for all \( x \) in the domain.

   Let's compute \( g(-x) \):
   \[
   g(-x) = (-x)^4 + 3(-x)^2 - 1
   \]

   Simplifying each term:
   \[
   (-x)^4 = x^4 \quad \text{(since raising to an even power makes it positive)}
   \]
   \[
   3(-x)^2 = 3x^2 \quad \text{(again, the even power makes it positive)}
   \]
   \[
   -1 \text{ remains the same.}
   \]

   Therefore:
   \[
   g(-x) = x^4 + 3x^2 - 1
   \]

   Since \( g(-x) = g(x) \), the function \( g(x) \) is **even**.

### 2. **Check for Oddness:**
   A function \( g(x) \) is **odd** if \( g(-x) = -g(x) \) for all \( x \) in the domain.

   We've already found that \( g(-x) = g(x) \), and for the function to be odd, \( g(-x) \) should equal \(-g(x)\). However:
   \[
   g(x) = x^4 + 3x^2 - 1
   \]
   \[
   -g(x) = -(x^4 + 3x^2 - 1) = -x^4 - 3x^2 + 1
   \]

   Clearly, \( g(-x) \neq -g(x) \). So, the function is **not odd**.

### **Conclusion:**
   The function \( g

Explore whether the function g(x) = x^4 + 3x^2 - 1 is even, odd, or neither with detailed step-by-step reasoning. This problem involves analyzing the properties of even and odd functions in polynomials, suitable for advanced high school students.

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