To determine the domain and parity of the function $g(x) = -x^2 + 3$:

Step 1: Determine the Domain

The function $g(x) = -x^2 + 3$ is a polynomial function, so it is defined for all real values of $x$. Therefore, the domain of $g(x)$ is: $D = \mathbb{R}$

Step 2: Determine the Parity

To check if $g(x)$ is even, odd, or neither, we evaluate $g(-x)$ and compare it with $g(x)$:

1. Compute $g(-x)$: $g(-x) = -(-x)^2 + 3 = -x^2 + 3 = g(x)$

Since $g(-x) = g(x)$, the function $g(x)$ is even. This means it is symmetric about the y-axis.

Would you like a deeper explanation, or do you have any specific questions?

Here are five related questions for further exploration:

1. What is the significance of even and odd functions in graph symmetry?
2. How would the parity change if the function were $g(x) = x^2 - 3$?
3. What happens to the symmetry if a constant is added to $g(x)$ that is not symmetric itself?
4. Can a function have both even and odd properties?
5. How does knowing the domain and range aid in graphing polynomial functions?

Tip: For any function, evaluating $f(-x)$ can help quickly determine if it’s even, odd, or neither.