To determine whether the function $f(x) = -x^2 - 1 + x^4$ is even, odd, or neither, we use the following definitions:

• A function is even if $f(-x) = f(x)$ for all $x$.
• A function is odd if $f(-x) = -f(x)$ for all $x$.
• If neither condition is satisfied, the function is neither even nor odd.

Step 1: Compute $f(-x)$

The given function is: $f(x) = -x^2 - 1 + x^4$

Substitute $-x$ into $f(x)$: $f(-x) = -(-x)^2 - 1 + (-x)^4$

Simplify: $f(-x) = -(x^2) - 1 + x^4$

$f(-x) = -x^2 - 1 + x^4$

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

From the computations above: $f(-x) = -x^2 - 1 + x^4$ $f(x) = -x^2 - 1 + x^4$

Since $f(-x) = f(x)$, the function satisfies the condition for being even.

Conclusion:

The function $f(x) = -x^2 - 1 + x^4$ is even.

---

Would you like further details or have any questions?

Expanding Questions:

1. How do you identify if a polynomial function is even, odd, or neither based on the exponents?
2. What does the symmetry of an even function imply geometrically?
3. Can a function be both even and odd? Why or why not?
4. Are all polynomials either even or odd?
5. What are some real-life applications of even functions?

Tip: For even functions, their graphs are symmetric about the y-axis.