Question

1. odd, even or neither. support your answer with algebra. y = -2x⁴ + 3x² + 3

Explanation:

Step1: Recall the definitions

A function \( y = f(x) \) is even if \( f(-x)=f(x) \) for all \( x \) in the domain, and odd if \( f(-x)= - f(x) \) for all \( x \) in the domain.

Step2: Compute \( f(-x) \)

Given \( f(x)=-2x^{4}+3x^{2}+3 \), substitute \( x \) with \( -x \):
\( f(-x)=-2(-x)^{4}+3(-x)^{2}+3 \)
Since \( (-x)^{n}=x^{n} \) when \( n \) is even, we have \( (-x)^{4}=x^{4} \) and \( (-x)^{2}=x^{2} \).
So \( f(-x)=-2x^{4}+3x^{2}+3 \)

Step3: Compare \( f(-x) \) and \( f(x) \)

We see that \( f(-x)=-2x^{4}+3x^{2}+3 = f(x) \)

Answer:

The function \( y = - 2x^{4}+3x^{2}+3 \) is an even function.