To determine if the function $f(x) = 3x^2 + 6$ is even, odd, or neither, we'll go through the steps given in the image.

Step 1: Evaluate $f(-x)$

We start by substituting $-x$ for $x$ in the function: $f(-x) = 3(-x)^2 + 6$ Simplify the expression: $f(-x) = 3x^2 + 6$

So, $f(-x) = 3x^2 + 6$.

Step 2: Compare $f(-x)$ with $f(x)$

We already know that: $f(x) = 3x^2 + 6$

Since $f(-x) = f(x)$, the function is even.

Final Conclusion

From the above work, we conclude that the function $f(x) = 3x^2 + 6$ is even.

Would you like more details on any specific step or do you have any questions?

Here are 5 related questions to further your understanding:

1. What is the general test for determining if a function is even?
2. How would the analysis differ if the function had an odd exponent?
3. Can a function be both even and odd? Explain.
4. How does the graph of an even function behave with respect to the y-axis?
5. How would you check if a function is neither even nor odd?

Tip: Remember, a function is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$.