To determine whether the function $p(x) = 3x^4 - 7x^2 + 9$ is even, odd, or neither, we can use the definitions of even and odd functions:

• A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$.
• A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$.

Step 1: Calculate $p(-x)$

We substitute $-x$ into the function and simplify:

$p(-x) = 3(-x)^4 - 7(-x)^2 + 9$

Simplifying each term:

• $(-x)^4 = x^4$, so $3(-x)^4 = 3x^4$,
• $(-x)^2 = x^2$, so $-7(-x)^2 = -7x^2$,
• The constant 9 remains unchanged.

Thus, we have:

$p(-x) = 3x^4 - 7x^2 + 9$

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

We can see that:

$p(-x) = 3x^4 - 7x^2 + 9 = p(x)$

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

Conclusion:

The function $p(x) = 3x^4 - 7x^2 + 9$ is even.

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. What is the difference between even and odd functions in terms of symmetry?
2. How would the function behave if there were an odd power of $x$?
3. Can a function be neither even nor odd? If so, how would you determine that?
4. How does the graph of an even function differ from that of an odd function?
5. What types of transformations can change the parity (even/odd nature) of a function?

Tip: An even function is always symmetric with respect to the y-axis, while an odd function is symmetric about the origin.